THE  UNIVERSITY 
OF  ILLINOIS 

LIBRARY 

From  the  collection  of 

Julius  Doerner,  Chicago 
Purchased,  1918. 

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lAILllOAD    ENGINEERS 


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FIE  LD-BOOK 


FOtt 


RAILROAD    ENGINEERS. 


CO.\TAI.MN(J 

F  0  R  M  U  L  /E 

IfOIi    LAYING    OUT    CURVES,    DETERMINING    FROG    ANGLES,     LEVELLING, 
CALCULATING    EARTH-WORK,    ETC.,    ETC., 

TOGETUER   WITH 

TABLES 

OF    )L\»II,   ORUINATE.S,    DEFLECTIONS,   LONG   CHORDS,   MAGNETIC  VAEIA 

TION,     LOGAKlTII.Mis,     LOGARITHMIC    AND     NATURAL    SINES, 

TANGENTS,    ETC.,   ETC. 


BY 

JOHN     B.    HENCK,   A.M., 

CIVIL    ENGINEER. 


NEW     YORK: 

D.  APPLETON   &    COMPANY,  549    &    551    BROADWAY. 

LONDON:    IG  LITTLE  BKITAIN 

1877. 


EsTEiiF-D,  according  to  Act  of  Congress,  in  the  year  1854, 

By  D.  APPLETON  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States 
for  the  Southern  District  of  Xow  York. 


6>Z  o  .  / 

/81^ 


PREFACE. 


The  object  of  the  present  work  is  to  supply  a  want  very 
i^enerally  felt  by  Assistant  Engineers  on  Railroads.  Books 
of  convenient  form  for  use  in  the  field,  containing  the  ordi 
nary  logarithmic  tables,  are  common  enough  ;  but  a  book 
combining  with  these  tables  others  peculiar  to  railroad 
work,  and  especially  the  necessary  formulse  for  laying  out 
curves,  turnouts,  crossings,  &c.,  is  yet  a  desideratum. 
These  formuke,  after  long  disuse  perhaps,  the  engineer  is 
often  called  upon  to  apply  at  a  moment's  notice  in  the 
field,  and  he  is,  therefore,  obliged  to  carry  with  him.  in 
manuscript  such  methods  as  he  has  been  able  to  mvent  or 
collect,  or  resort  to  what  has  received  the  very  appropriate 
name  of  "  fudging."  This  the  intelligent  engineer  always 
considers  a  reproach;  and  he  will,  therefore,  it  is  hoped, 
receive  with  favor  any  attempt  to  make  a  resort  to  it  inex- 
cusable. 

Besides  supplying  the  want  just  alluded  to,  it  was  thought 
that  some  improvements  upon  former  methods  might  be 
made,  and  some  entirely  new  methods  introduced.  Among 
the  processes  believed  to  be  original  may  be  specified 
those  in  §§41  — 48,  on  Compound  Curves,  m  Chapter  II., 
on  Parabolic  Curves,  in  §§  106  -  109,  on  Vertical  Curves, 
and  in  the  article  on  Excavation   and   Embankment.     It  is 

4694 4? 


V]  PREFACE. 

but  just  to  add,  that  a  great  part  of  what  is  said  on  Reversed 
Curves,  Turnouts,  and  Crossings,  and  most  of  the  Miscel- 
laneous Problems,  are  the  result  of  original  investigations. 
In  the  remaining  portions,  also,  many  simplifications  have 
been  made.  In  all  parts  the  object  has  been  to  reduce  the 
operation  necessary  in  the  field  to  a  single  process,  inil;- 
cated  by  a  formula  standing  on  a  line  by  itself,  and  distin- 
guished by  a  ly .  This  could  not  be  done  in  all  cases,  as 
will  be  readily  seen  on  examination.  Certain  preliminary 
steps  were  sometimes  necessary,  and  these,  whenever  it 
was  practicable,  have  been  indicated  by  words  in  italics. 

Of  the  methods  given  for  Compound  Curves,  that  in 
§  46  will  be  found  particularly  useful,  from  the  great  variety 
of  applications  of  which  it  is  susceptible. 

Methods  of  laying  out  Parabolic  Cui-ves  are  here  given, 
that  those  so  disposed  may  test  their  reputed  advantages. 
Two  things  are  certainly  in  their  favor ;  they  are  adapted 
to  unequal  as  well  as  equal  tangents,  and  their  cuiTature 
generally  decreases  tov/ards  both  extremities,  thus  making 
the  transition  to  and  from  a  straio-ht  line  easier.  Some 
labor  has  been  given  to  devising  convenient  ways  of  laying 
out  these  curves.  The  method  of  determinins;  the  radius 
of  curvature  at  certain  points  is  believed  to  be  entirely 
WQW.  Better  processes,  however,  may  already  exist,  par- 
ticularly in  France,  where  these  curves  are  said  to  be  in 
general  use. 

The  mode  of  calculating  Excavation  and  Embankment 
here  presented,  will,  it  is  thought,  be  found  at  least  as  sim- 
ple and  expeditious  as  those  commonly  used,  with  the  ad- 
vantage over  most  of  them  in  point  of  accuracy.  The  usual 
Tables  of  Excavation  and  Embankment  have  been  omitted. 
To  include  all  the  varieties  of  slope,  width  of  road-b^d,  and 
depth  of  cuttmg,  they  must  be  of  great  extent,  and  uiitiued 

H 


PREFACE.  ri:, 

tor  a  field-book.  Even  then  they  apply  only  to  ground 
whose  cross-section  is  level,  though  often  used  in  a  mannei 
shown  to  be  erroneous  in  §  128.  When  the  cross-section 
of  the  ground  is  level,  the  place  of  the  tables  is  supplied  by 
the  formula  of  §  119,  and  when  several  sections  are  calcu- 
lated together,  as  is  usually  the  case,  and  the  work  is  ar- 
ranged in  tabular  form,  as  in  §  120,  the  calculation  is  be- 
lieved to  be  at  least  as  short  as  by  the  most  extended  tables. 
The  correction  in  excavation  on  curves  (§  129)  is  not 
known  to  have  been  introduced  elsewhere. 

In  a  work  of  this  kind,  brevity  is  an  essential  feature. 
The  form  of  "Problem"  and  "Solution"  has,  therefore, 
been  adopted,  as  presenting  most  concisely  the  thing  to  be 
done  and  the  manner  of  doing  it.  Every  solution,  how- 
ever, carries  with  it  a  demonstration,  which  is  deemed  an 
equally  essential  feature.  These  demonstrations,  with  a 
few  unavoidable  exceptions,  principally  in  Chapter  II.,  pre- 
suppose a  knowledge  of  nothing  beyond  Algebra,  Geome- 
try, and  Trigonometry.  The  result  is  in  general  expressed 
by  an  algebraic  formula,  and  not  in  words.  Those  familiar 
with  algebraic  symbols  need  not  Jje  told  how  much  more 
uitelligible  and  quickly  apprehended  a  process  becomes 
when  thus  expressed.  Those  not  familiar  with  these  sym- 
bols should  lose  no  time  in  acquiring  the  ready  use  of  a 
language  so  direct  and  expressive.  It  may  be  remarked 
that  it  was  no  part  of  the  author's  design  to  furnish  a  col- 
lection of  mere  "  rules,"  professing  to  require  only  an  abil- 
ity to  read  for  their  successful  application.  Rules  can  sel- 
iom  be  safely  applied  without  a  thorough  understanding  of 
llie  principles  on  which  they  rest,  and  such  an  understand- 
ing, in  the  present  case,  implies  a  knowledge  of  algebraic 
(ormulse. 

The  tables  here  presented  will,  it  is  hoped,  prove  relia 


VUl  PREFACE. 

ble.  Those  specially  prepared  for  this  work  have  been 
computed  with  great  care.  The  values  have  in  some  cases 
been  carried  out  farther  than  ordinary  practice  requires,  in 
order  that  interpolated  values  may  be  obtained  from  them 
more  accurately.  For  the  greater  part  of  the  material 
composing  the  Table  of  Magnetic  Variation  the  author  is 
indebted  to  Professor  Bache,  whose  distinguished  ability  ir 
conducting  the  operations  of  the  Coast  Survey  is  equalled 
only  by  iiis  desire  to  diffuse  its  results.  The  remaining 
tables  have  been  carefully  examined  by  comparing  them 
with  others  of  approved  reputation  for  accuracy.  Many 
errors  have  in  this  way  been  detected  in  some  of  the  tables 
of  corresponding  extent  in  general  use,  particularly  in  the 
Table  of  Squares,  Cubes,  &c.,  and  the  Tables  of  Logarith- 
mic and  Natural  Sines,  Cosines,  &c.  The  number  of  tables 
might  have  been  greatly  increased,  but  for  an  unwillingness 
to  insert  any  thing  not  falling  strictly  within  the  plan  of  th? 
work  or  not  resting  on  sufficient  authority. 

J.  B.  11. 

Boston,  February,  1854. 


TABLE    OF    CONTENTS. 


CHAPTER  I. 

CIRCULAR    CURVES. 

Article  I.  —  Simple  Curves. 

2.  Definitions.     Propositions  relating  to  the  circle       .        .  1 

4.  Angle  of  intersection  and  radius  given,  to  find  the  tangent  3 

5.  Angle  of  intersection  and  tangent  given,  to  find  the  radius  3 

6.  Degree  of  a  curve 4 

7.  Deflection  angle  of  a  curve ♦ 

A.     Method  by  Deflection  Angles. 

9.  Radius  given,  to  find  the  deflection  angle         ....       4 

10.  Deflection  angle  given,  to  find  the  radius  .         .         ,  4 

11.  Angle  of  intersection  and  tangent  given,  to  find  the  deflection 

angle        .  5 

12.  Angle  of  intersection  and  deflection  angle  given,  to  find  the 


tangent 


13    Angle  of  intersection  and  deflection  angle  given,  to  find  the 

length  of  the  curve 6 

U.  Deflection  angle  given,  to  lay  out  a  curve        ....  7 

.16.  To  find  a  tangent  at  any  station 8 

B.     Method  1)1/  Tangent  and  Chord  Dejlections. 

17.  Definitions    ...  ....  .8 

18.  Radius  given,  to  find  the  tangent  deflection  and  chord  deflection  9 

19.  Deflection  angle  given,  to  find  the  chord  deflection      .        .  9 

21.  To  find  a  tangent  at  any  station 9 

22.  Chord  deflection  given,  to  lay  out  a  curve    .        .        .         .  10 


S  TABLE    OF    CONTENTS. 

C.     Ordinatcs. 

24.  Definition •  •        •         H 

25.  Deflection  angle  or  radius  given,  to  find  ordinatcs  .     11 

26.  Approximate  value  for  middle  ordinate        .        .  .        ■         l-^ 

27.  Method  of  finding  intermediate  points  on  a  curve  approxi- 

mately         .                         .                 •  .        .     14 

D.     Cui~ving  Rails. 

29.  Deflection  angle  or  radius  given,  to  find  the  ordinate  for  curv- 

ing rails        .         •  ^'^ 

Article  II.  — Reveesed  and  Compound  Ccrtes, 

30.  Definitions •        •        •        .15 

31.  Radii  or  deflection  angles  given,  to  lav  out  a  reversed  or  com- 

pound curve ^^ 

A.  Reversed  Curves. 

32.  Reversing  point  when  the  tangents  are  parallel       .        .  16 

33.  To  find  the  common  radius  when  the  tangents  are  parallel  1 6 

34.  One  radius  given,  to  find  the  other  when  the  tangents  are  par- 

allel  ....  " 

35.  Chords  given,  to  find  the  radii  when  the  tangents  are  parallel  18 

36.  Radii  given,  to  find  the  chords  when  the  tangents  are  parallel  18 

37.  Common  radius  given,  to  run  the  curve  when  the  tangents  are 

not  parallel ^^ 

38.  One  radius  given,  to  find  the  other  when  the  tangents  are  not 

parallel *^ 

39.  To  find  the  common  radius  when  the  tangents  are  not  parallel    21 

40.  Second  method  of  finding  the  common  radius  when  the  tan- 

gents are  not  parallel 22 

B.  Compound  Curves. 

41.  Common  tangent  point    ....  .23 

42.  To  find  a  limit  in  one  direction  of  each  radius     .  .        24 

44.  One  radius  given,  to  find  the  other 25 

45.  Second  method  of  finding  one  radius  when  the  other  is  given  26 

46.  To  find  the  two  radii 2V 

47.  To  find  the  tangents  of  the  two  branches      ....  29 
48    Second  method  of  finding  the  tangents  of  the  two  branches     .  30 


TABLE    OF    CONTENTS.  B 

Article  III.  —  Turxouts  and  Crossings. 

HECT.  PAQl 

i9.  Dcliiiitions '^1 

A.     Turnout  from  Straight  Lines. 

50.  Radius  given,  to  find  the  frog  angle  and  the  position  of  the  frog  32 

51.  Frog  angle  given,  to  find  the  radius  and  the  position  of  the  frog  33 

52.  To  find  mechanically  the  proper  position  of  a  given  frog        .  34 

53.  Turnouts  that  reverse  and  become  parallel  to  the  main  track  34 

54.  To  find  the  second  radius  of  a  turnout  reversing  opposite  the 

frog  .......                 ...  35 

B,     Crossings  on  Straight  Lines. 

55.  Kcferences  to  proper  problems 36 

56.  Radii  given,  to  find  the  distance  between  switches            .  36 

C.     Turnout  from  Curves. 

57.  Frog  angle  given,  to  find  the  radius  and  the  position  of  the  frog  38 

58  To  find  mechanically  the  proper  position  of  a  given  frog     .  41 

59  Proper  angle  for  frogs  that  they  may  come  at  the  end  of  a  rail  41 

60  Radius  given,  to  find  the  frog  angle  and  the  position  of  the  frog  42 
62    Turnout  to  reverse  and  become  parallel  to  the  main  track.       .  44 

D.     Crossings  on  Curves. 

63.  References  to  proper  problems •  ^^ 

64.  Common  radius  given,  to  find  the  central  angles  and  chords  45 

Article  IV.  —  Miscellaneous  Problems. 

65.  To  find  the  radius  of  a  curve  to  pass  through  a  given  point  46 

66.  To  find  the  tangent  point  of  a  curve  to  pass  through  a  given 

point 47 

67.  To  find  the  distance  to  the  curve  from  any  point  on  the  tan- 

gent    47 

68    Second  method  for  passing  a  curve  through  a  given  point       .  47 

69.  To  find  the  proper  chord  for  any  angle  of  deflection    .        .  4* 

70.  To  find  the  radius  when  the  distance  from  the  intersection 

point  to  the  curve  is  given 48 

71    To  find  the  distance  from  the  intersection  point  to  the  curve 

when  the  radius  is  given  ...                ...  49 


Xll  TABLE    OF    CONTENTS. 

SECT.  PAai 

72.  To  finil  the  ta\igent  point  of  a  curve  that  shall  pass  through  a 

given  point  ....  .  5C 

73.  To  find  the  radius  of  a  curve  without  measuring:  angles      .         51 

74.  To  find  the  tangent  points  of  a  curve  without  measuring  an- 

gles .        ,  .  ...         5? 

75.  To  find  the  angle  of  intersection  and  the  tangent  points  when 

the  point  of  intersection  is  inaccessible     ....        52 

76.  To  lay  out  a  curve  when  obstructions  occur  .         .     5.t 

77.  To  change  the  tangent  point  of  a  curve,  so  that  it  may  pass 

through  a  given  pomt 50 

78.  To  change  the  radius  of  a  curve,  so  that  it  may  terminate  in 

a  tangent  parallel  to  its  present  tangent      .        .        .        .57 

79.  To  find  the  radius  of  a  curve  on  a  track  alreadv  laid  .        .        5;^ 

80.  To  draw  a  tangent  to  a  given  curve  from  a  given  point  .        .     59 

81.  To  flatten  the  extremities  of  a  sharp  curve  ....         .tj 

82.  To  locate  a  curve  without  setting  the  instrument  at  the  tan- 

gent point         .         .  .  ....         60 

'*.'?.  To  measure  the  distance  across  a  river  .  6.H 


CHAPTER    II. 

PARABOLIC    CURVES. 

Article  I.  —  Locating  Parabolic  Clkvls. 

84.  Fropo.>itions  relating  to  the  parabola       ...  .65 

85.  To  lay  out  a  parabola  by  tangent  deflections  ...  66 
36.  To  lay  out  a  parabola  by  middle  ordinates  .  .  .  .67 
87.  To  draw  a  tangent  to  a  parabola 67 

89.  To  lay  out  a  parabola  by  bisecting  tangents     •         -         .  .68 

90.  To  Iny  out  a  parabola  by  intersections          ...  69 

Ai;tict.e  II. —  Radius  of  Curvature. 

9^.  Definition      ....  ...  .71 

9-3.  To  find  the  radius  of  curvature  at  certain  stations    .         .  .71 

95.  Simplification  when  the  tangents  are  equal          .        .        .  7« 


TABLE    OF    CONTENTS.  XIH 


CHAPTER    III. 
LEVELLING. 

AnriCLE  I.  —  Heights  and  Slope  Stakes. 

»»JT.  PAGE 

96.  Definitions 78 

97.  To  find  the  diflovence  of  level  of  two  points   .         .        .  .78 

98    Datum  plane 79 

99.  To  find  tlic  heights  of  the  stations  on  a  line   .        .         .  .     8C 

100.  Sights  denominated  jo/ms  and  m««Ms 81 

101.  Form  of  field  notes 82 

102.  To  set  slope  stakes 82 

AuTiCLE  II.  —  Correction  for  the  Earth's  Curvature  and 

FOR  Refraction. 

103.  Earth's  curvature 84 

104.  Refraction 84 

105.  To  find  tlie  correction  for  curvature  and  refraction        .         .     85 

Article  III.  —  Vertical  Curves. 

106.  Manner  of  designating  grades         .  86 

107.  To  find  the  grades  for  a  vertical  curve  at  whole  stations  86 

109.  To  find  tlie  grades  for  a  vertical  curve  at  sub-stations    .  88 

Article  [V. —  Elevation  of  the  Outer  Rail  on  Curves. 

110.  To  find  the  proper  elevation  of  the  outer  rail  89 
.11.  Coning  of  the  wheels 89 


CHAPTER   IV. 

EARTII-WORK, 
Article  I.  —  Prismoidal  For.mula. 

.12    Definition  of  a  prismoid 92 

[13.  To  find  the  solidity  of  a  piismoid 92 

Article  II    -Borroav-Pits. 
114.  Manner  of  dividing  the  ground 93 


XIV  T..i5LE    OF    CONTENTS. 

SECT.  PAOa 

115.  To  find  the  solidity  of  a  vertical  prism  whose  horizontal  sec- 

tion is  a  triangle 93 

116.  To  find  the  solidity  of  a  vertical  prism  whose  horizontal  sec- 

tion is  a  parallelogram 94 

117.  To  find  the  solidity  of  a  number  of  adjacent  prisms  having 

the  same  horizontal  section f '^ 


\rticle  III.  —  Excavation  and  Embankment. 
A.     Centre  Heights  alone  given. 

119.  To  find  the  solidity  of  one  section 97 

120.  To  find  the  solidity  of  any  number  of  successive  sections       .    98 

B.     Centre  and  Side  Heights  given. 

121.  Mode  of  dividing  the  ground 9^ 

122.  To  find  the  solidity  of  one  section lUO 

123.  To  find  the  solidity  of  any  number  of  successive  sections       .  104 

125.  To  find  the  solidity  when  the  section  is  partly  in  excavation 

and  partly  in  embankment  ....  .        .  105 

126.  Beginning  and  end  of  an  excavation  ...  .       107 

C.     Ground  very  Irregular. 

127.  To  find  the  solidity  when  the  ground  is  very  irregular       .       108 

128.  Usual  modes  of  calculating  excavation 109 

D.     Correction  in  Excavation  on  Curves. 

129.  Nature  of  the  correction 110 

130.  To  find  the  correction  in  excavation  on  curves  .        .        .       112 
132.  To  find  the  correction  when  the  section  is  partly  in  excava 

tion  and  partly  in  embankment -113 


TABLES. 

RO.  PAOB 

I.    Radii,  Ordinates,  Tangent  and  Chord  Deflections,  and  Or- 

dinates  for  Curving  Rails 115 

U.    Long  Chords 119 


TABLE    OP    CONTENTS.  X^ 

NO.  PAGE 

HI.  (correction  for  the  Earth's  Curvature  and  for  Rcfract'uin    .  120 

IV.  Elevation  of  the  Outer  Rail  on  Curves   .         .        .        .       I'iO 

V.  Frog  Angles,  Chords,  and  Ordinates  for  Turnouts    .        .121 

VI.  Length  of  Circular  Arcs  in  Parts  of  Radius  .         .        .       121 

VJI.     Expansion  by  Heat 122 

VIII.     Properties  of  Materials 123 

IX.     Magnetic  Variation 126 

X.  Trigonometrical  and  Miscellaneous  Ft  (-mulie        .         .       13'i 

XI  Squares,  Cubes,  Square  Roots,  Cube  Roots,  and  Recip- 
rocals .......                 .        .       137 

XII.  Log Arithms  of  Numbers      .        .                 ....  155 

XIII.  Logarithmic  Sines,  Cosinee  Tangents,  and  Cotangents        171 

XIV.  Natural  Sines  and  Cosines 219 

XV.  Natural  Tangents  and  Cotangents          .        .                .      229 

XVL  Rise  per  Mile  of  Various  Grades       ....          MJ 


EXPLANATION    OF    SIGNS. 


The  sign  +  indicates  that  the  quantities  between  which  it  is  placed 
ire  to  be  added  together. 

The  sign  —  indicates  that  the  quantity  before  which  it  is  placed 
.s  to  be  subtracted. 

The  sign  X  indicates  tliat  the  juantities  between  which  it  is  placed 
are  to  be  midtiplied  together. 

The  sign  -r-  or  :  indicates  that  the  fust  of  two  quantities  between 
which  it  is  placed  is  to  be  divided  by  the  second. 

The  sign  —  indicates  that  the  quantities  between  which  it  is  placed 
are  equal. 

The  sign  oo   indicates  that  the  difference  of  the  two  quantities  be- 
tween which  it  is  placed  is  to  be  taken 

The  sign  .• .  stands  for  the  word  "hence  "  or  "  therefore." 

The  ratio  of  one  quantity  to  another  may  be  regarded  as  the  quo- 
tient of  the  first  divided  by  the  second.     Hence,  the  ratio  of  a  to  6  is 
expressed  by  a  :  h,  and   the  ratio  of  c  to  d  by  c  :  (/.     A  proportion  ex 
presses  tlie  equal  it  1/  of  two  latios.     Hence,  .  proportion  is  rcjiresented 
by  placing  the  sign  —  between  two  ratios  ;  as,  a  ■  b  =  c  :  d 


In  the  text  and  in  the  tables  the  foot  has  been  taken  as  the  unit  gi 
measure  when  no  other  unit  is  specified. 


FIELD-BOOK. 


CH/VPTER    I. 


CIRCULAR   CURVES. 


Article  I.  —  Simple  Cuka'es 


1.  The  railroad  curves  here  considered  are  eitlier  Circular  or  Para 
holic.  Circular  curves  are  divided  into  Simple,  Reversed,  and  Com 
j)Ound  Curves.     We  begin  with  Simple  Curves. 

2.  Let  the  arc  ADEFB  (fig.  1)  represent  a  railroad  ciu've,  unit 


Fig.  \. 


2  CIRCULAR    CURVES. 

ing  the  straight  lines  GA  and  B  FT.  The  lengtli  of  sudi  a  curve  is 
measured  by  cliords,  each  100  feet  long.*  Tlius,  if  the  chords  AD^ 
DE,  E  F,  and  FB  are  each  100  feet  in  length,  the  whole  curve  is 
said  to  be  400  feet  long.  The  straight  lines  GA  and  BH  are  always 
tangent  to  the  curve  at  its  extremities,  which  are  called  tangent  points. 
U  GA  and  BH  are  produced,  until  they  meet  in  C,  ^  C  and  B  C 
are  called  the  tangents  of  the  curve.  If  ^  C  is  produced  a  little  beyond 
Cto  /v,  the  angle  KGB,  formed  by  one  tangent  with  the  other  pro- 
duced, is  called  the  angle  of  intersection,  and  shows  the  change  of  direc- 
tion in  passing  from  one  tangent  to  the  other. 

The  following  propositions  relating  to  the  circle  are  derived  from 
Geometry. 

I.  A  tangent  to  a  circle  is  perpendicular  to  the  radius  drawn  through 
the  tangent  point.  Thus,  A  C  is  perpendicular  to  A  0,  and  B  C  to 
BO. 

II.  Two  tangents  drawn  to  a  circle  from  any  point  are  equal,  and  it 
a  chord  be  drawn  between  the  two  tangent  points,  the  angles  between 
this  chord  and  the  tangents  are  equal.  Thus  AC—  B  C,  and  the 
angle  B  A  C  =^  A  B  C. 

III.  An  acute  angle  between  a  tangent  and  a  chord  is  equal  to  half 
the  central  angle  subtended  by  the  same  chord.  Thus,  C  A  B  — 
hAOB. 

IV.  An  acute  angle  subtended  by  a  chord,  and  having  its  vertex  in 
the  circumference  of  a  circle,  is  equal  to  half  the  central  angle  sub- 
tended by  the  same  chord.     Thus,  D  AE  =  i  D  OE. 

V.  Equal  chords  subtend  equal  angles  at  the  centre  of  a  circle,  and 
also  at  the  circumference,  if  the  angles  are  inscribed  in  similar  seg- 
ments.    Thus,  AOD  =  DOE,  and  D  A  E  =  E  A  F. 

VI.  The  angle  of  intersection  of  two  tangents  is  equal  to  the  cen- 
tral angle  subtended  by  the  chord  which  unites  the  tangent  points. 
Thus,  KGB  =  AO b' 

3.  In  order  to  unite  two  straight  lines,  as  GA  and  B H,  by  a  curve, 
the  angle  of  intersection  is  measured,  and  then  a  radius  for  the  curve 
may  be  assumed,  and  the  tangents  calculated,  or  the  tangents  may  be 
assumed  of  a  certain  length,  and  the  radius  calculated. 


*  Some  engineers  prefer  a  chain  50  feet  in  length,  and  measui'e  the  length  cf  :i 
enrve  by  chords  of  50  instead  of  100  feet.  The  chord  of  100  feet  has  been  adopteii 
throughout  this  article ;  but  the  formulae  deduced  may  be  very  readily  modified  t(. 
Buit  chords  of  any  length.     See  also  ^  13. 


SIMPLE    CURVES. 


ti 


4.    Pro'bleni.     Given  the  angle  of  intersection  K  C  B  —  1  fjig   \) 
and  the  radius  A  0  =  R,  tojind  the  tangent  A  C  =  T. 


1-iy  I. 


Solution.     ])niw   CO.     Then  in  the  right  triangle  AOC  we  lia«'', 

iTab.  X.  3)    4-;-  =  tan.  AO  C,  or,  since  A  0  0=^1  a  2,  VI.) 
A  O 


-  =  tan.  2  /; 


T  =  R  tan.  ^  /. 


Example.     Given  7  =  22==  .52',  and  /?  =  3000,  to  find  T.     Here 

A' =  3000  3.477121 

^7=11°  26'  tan.  9.305865 


T=  606  72 


2.7829»0 


.5.    Problem.      Given  the  angle  of  intersection  KCB  =  I  {fg.  I ), 
ind  the  tangent  A  C  —   '1\  to  find  thp  radius  A  0  =-.  R. 


4  CIRCULAR    CURVES. 

Solution.      In    tlie   right    triangle   A  0  C   we    have    (Tab.    X.   61 

—  =  cot.  A  O  C.  ov   —  =  cot.  h  i ; 
AC  '        r  ^    ' 

!^=  ,'.  R==  Tcot.  i/. 

Example.     Given  7  =  31°  16'  and   r=  950,  to  find  72.     Here 

r=950  2.977724 

^1=  15°  38  cot.  0.553102 


R  =  3394.89  3.530826 

6.  The  decree  of  a  curve  is  determined  by  the  angle  subtended  at 
its  centre  by  a  chord  of  100  feet.  Thus,  if  A  0  D  =  6°  (fig.  1), 
ADEFB  is  a  6°  curve. 

7.  Tlie  deflection  angle  of  a  curve  is  the  acute  angle  formed  at  any 
point  between  a  tangent  and  a  chord  of  100  feet.  The  deflection  angle 
is,  therefore  (^  2,  III ),  half  the  degree  of  the  curve.  Thus,  CAD  or 
CBF  is  the  deflection  angle  of  the  curve  A  D  E  F B,  and  is  half 
A  OD  or  half  F 0  B. 


A.    Method  by  Deflection  Angles. 

8.  The  usual  method  of  laying  out  a  curve  on  the  ground  is  by 
means  of  deflection  angles. 

9.  Problem.      Given  the  radius  A  0  ==  R  {fig.  \),  to  find  the  de- 
flection angle  C  B  F  =  D. 

Solution.    Draw  OL  perpendicular  to  B  F.     Then  the  angle  BOL 
=  hBOF=  D,  and  BL  =  hBF=50.     But  in  the  right  triangle 

OBL  yve  have  (Tab.  X.  1 )  sin.  BOL  =  ^; 
IW  sin.  Z)  =  — . 

J.  L 

Example.     Given  R  =  5729.65,  to  find  D.     Here 

50  1.698970 

72  =  5729.65  3.758128 


D  =  30'  sin.  7.940842     . 

Hence  a  curve  of  this  radius  is  a  1°  curve,  and  its  deflection  angle  is 
30'. 

10.    Problem.     Given  the  deflection  angle  C B  F  =  D  (fig.  1),  «» 

find  the  radius  A  0  ^=  R.      ■ 


METHOD    BY    DEFLECTION    ANGLES.  5 

Solution.    By  the  preceding  section  we  have  sin.  Z)=  — ,  whence 

R 

fi  sin.  D=^  50; 

50 


'.  A'  = 


sin.  D 
By  this  formula  the  radii  in  Tahle  I.  are  calculated. 

Erampk.     Given  D  =  1",  to  find  R.      Here 

50  1.698970 

■^=1''  sin.  8  241S.')5 


i^=  2864.93  3.457115 

1 1 .    Problem.     Given  the  angle  of  intersection  KCB  =  I  (Jig.  1 ), 
and  the  tangent  AC  =  T,  to  find  the  deflection  angle  CA  D  =  D. 

Solution.    From   §  9    we   have   sin.  D  =  —,  and  from   ^  5,  R  = 

7' cot.  .^7.     Substituting  this  value  of  72  iv  the  first  equation,  we  get 

sm.  D  = ; 

rcot.  i  /' 

r5s«                                     •      T-.       50  tan.  i  / 
ts^  .  • .  sm.  D  = L_  . 


Example.     Given  7  =  21°  and  T  =  424.8,  to  find  D.     Here 

50  1.698970 

^7=10°  30  tan.  9.267967 


0.9669S7 
7' =424  8  2.628185 


7)  =  1°  15'  sin.  8.338752 

12.    Problem.     Given  the  angle  of  intersection  KCB  ^  I  {fig.  \) 
and  the  deflection  angle  CAD  =  D,  to  find  the  tangent  AC=  T. 

Solution.    From  the  preceding  section  we  have  sin.  D  =  -  ^°'  ^-\ 

T 
Hence,  Tsin.  7)  =  50  tan.  i  7; 

j^=»  .     rp 50  tan.  «i  7 

sin.  D 

Example.     Given  7  =  28°  and  D  =  1°,  to  find  T.    Here 

„       50  tan.  14° 

T=  -~r~Tr-  =  714.31. 
Bin  l"" 


b  CIRCULAR    CURVES. 

13.  Problem.  Given  the  angle  of  intersei  tion  K  CB  =  I  {Juf.  1), 
and  the  deflection  angle  C  A  D  =  D,  to  find  the  length  of  the  curve. 

Solution.  By  §  2  the  length  of  a  curve  is  measured  by  chords  of  100 
feet  applied  around  the  curve.  Now  the  first  chord  A  D  makes  with 
the  tangent  A  C  oxi  angle  C A  D  =^  D,  and  each  succeeding  chord 
DE,EF,&c.  subtends  at  u4  an  additional  angle  DAE,  EAF,  &c. 
each  equal  to  D;  since  each  of  these  angles  (§  2,  IV.)  is  half  of  a 
central  angle  subtended  by  a  chord  of  100  feet.  The  angle  CAB  = 
i  A  0  B  =  ^  I  is,  therefore,  made  up  of  as  many  times  Z),  as  there  are 
chords  around  the  curve.  Then  if  n  represents  the  number  of  chords, 
we  have  n  D  =  ^  I', 

hi 

,• .  n  =  - — . 

D 

If  D  is  not  contained  an  even  number  of  times  in  ^  /,  the  quotient 
above  will  still  give  the  length  of  the  curve.  Thus,  in  fig.  2,  suppose 
D  is  contained  4|  times  in  ^  /.  This  shows  that  there  will  be  four 
whole  chords  and  |  of  a  chord  around  the  curve  from  A  to  B.  The 
angle  GAB,  the  fraction  of  D,  is  called  a  sub  deflection  angle,  and 
G  B.  the  fraction  of  a  choi'd,  is  called  a  sub-chord* 

The  length  of  the  curve  thus  found  is  not  the  actual  length  of  tlie 
arc,  but  the  length  required  in  locating  a  curve.  If  the  actual  length 
of  the  arc  is  required,  it  may  be  found  by  means  of  Table  VI. 

Example.     Given  /  =  16°  52'  and  D  =  \°  20',  to  find  the  length  of 

JL   J-  g3  9gl  506' 

the  curve.     Here   n  =  '^  =  £5^  "=80^""  ^•^-^'  ^^^^  ^^'  *®  ^"^^® 
is  6.32.5  feet  long. 

To  find  the  arc  itself  in  this  example,  we  take  from  Table  VI.  the 
length  of  an  arc  of  I60  52',  since  the  central  angle  of  the  whole  curve 
is  equal  to  /(§  2,  VI  ),  and  multiply  this  length  by  the  radius  of  the 

curve. 

Arc  10°  =  .1745329 

"   6°  =  .1047198 

«  50'  =  .0145444 

«   2'  =  .0005818 


"  16°  52'  =  .2943789 


•  This  method  of  finding  the  length  of  a  sub-chord  is  not  mathematically  accu- 
rate ;  for,  by  geometry,  angles  inscribed  in  a  circle  are  proportional  to  the  arcs  on 
which  they  stand ;  whereas  this  method  supposes  them  to  be  proportional  to  the 
chords  of  these  arcs.  lu  railroad  curves,  the  error  arising  from  this  supposition  m 
too  small  to  be  regari'ed. 


METHOD    BY    TiiniENT    AND   CHORD    DEFLECTIONS.  9 

o»rt%  B  11  and  C  K  of  tlie  same  length  as  the  chords.  Draw  O/i 
nnd  D  K.  B  G  is  called  the  tangent  deflection,  and  C  H  or  D  K  the 
du>nl  deflection. 

18.  Problem.  Given  the  radius  AO  =  R  (flg.  S),  to  flnd  the 
tangent  deflection  B  G,  and  die  chord  deflection  C  H. 

Solution.  The  triangle  C  B  II  is  similar  to  BOC;  for*thc  angle 
BOC=  180=  -  {OBC-\-  B  CO),  or,  since  BCO  =  ABO,  BOC 
=  180=  —  {0  BC  -{-  ABO)  =  CB  H,  and,  as  both  tiie  triangles  are 
isosceles,  the  remaining  angles  are  equal.  The  homologous  sides  are. 
therefore,  proportional,  that  is,  B  0  :  B  C  =  B  C :  C II,  or,  represent- 
ing, the  chord  by  c  and  the  chord  deflection  by  d,  R  :  c  =^  c  :  d\ 

c^ 

^  .-.  d  =  -. 

R 

To  find  the  tangent  deflection,  draw  BM  to  the  middle  of  6*7/, 
bisecting  the  angle  C B  H,  and  making  i3il/C  a  right  angle.  Then 
the  right  triangles  B  M  C  and  AGS  are  equal ;  fovBC=A  B,  and 
the  angle  CBM=hCBII=iBOC=^AOB  =  BAG  (§2, 
III.).  Therefore  B  G  =  CM=  h  OH  =  ^d,  that  is,  the  tangent  de- 
flection is  half  the  chord  deflection. 

19.  Pr61>!eill.  Given  the  deflection  angle  D  of  a  curve,  to  flnd  the 
chord  deflection  d. 

Solution.     By  the  precedin;;  section  we  have  d^=  -77,  and  by  \  10, 

tl  =   , — ^      Substituting  this  value  of  R  in  the  first  equation,  we  find 

c^  sin.  D 


^  d  = 


50 

This  formula  gives  the  chord  deflection  for  a  chord  c  of  any  length 
though  D  is  the  deflection  angle  for  a  chord  of  100  feet  (^^  7).  When 
c  =  100,  the  formula  becomes  d=  200  sin  D,  or  for  the  tangent  de- 
flection hd  =  100  sin.  D.  By  these  formulte  the  tangent  and  chord 
deflections  in  Table  I.  may  be  easily  obtained  from  the  table  of  natural 
sines 

20.  The  length  of  the  curve  may  be  found  by  first  finding  Z)  (§  9  or 
J  U),  and  then  proceeding  as  in  §  13. 

21.  Probleifla      To  drcntJ  a  tangent   to  the  cun^e  at  any  station, 
HS  B  {Jig.  3). 

Solution.    Bisect  tne  chord  deflection  II 0  of  the  next  station  in  M. 
2 


10  CIRCULAR    CURVES. 

A  line  drawn  through  B  and  31  will  be  the  tangent  required ;  foi  it 
has  been  proved  (§  18)  that  the  angle  C  B  M  is  in  this  case  equai  to 
i  B  0  0,  and  B  J/ is  consequently  (§  2,  III.)  a  tangent  at  B. 

If  B  is  at  the  end  of  the  curve,  the  tangent  at  B  may  be  found  with- 
out first  laying  off  //  C.  Thus,  if  a  chain  equal  to  the  chord  is  extend- 
ed to  H  on  A  B  produced,  the  point  H  marked,  and  the  chain  ihon 
swung  ronnd,  keeping  the  end  at  B  fixed,  until  II M  =  h  d,  fJ  M  will 
he  the  direction  of  the  re(iuired  tangent.* 

22.  ProtoleilS.  Giveii  the  chord  deflection  (/,  to  lay  .nil  a  curcc 
from  a  given  tangent  point. 

Solution.     Let  A  (tig.  3)  be  the  given  tangent  point,  and  suppose  '/ 
has  been  calculated  for  a  chord  of  100  feet.     Stretch  a  cbain  of  li'i; 
feet  from  A  to  G  on  the  tangent  EA  produced,  and  mark  the  poini 
G.     Swing  the  chain  round  towards  AB,  keeping  the  end  at  A  fixed 
until  B  G  \s  equal  to  the  tangent  deflection  i  c/,  and  B  will  be  the  first 
station  on  the  curve.     Stretch  the  chain  from  B  to  H  on  AB  pro 
duced,  and  having  marked  this  point,  swing  the  chain  round,  until  U  C 
is  etpial  to  the  chord  deflection  d.     Cis  the  second  station  on  the  curve 
Continue  to  lay  off  the  chord  deflection  from  the  preceding  chord  pro 
duced,  until  the  curve  is  finished. 

Should  a  sub-chord  DF  occur  at  the  end  of  the  curve,  find  the  tan 
gent  DL  at  D  (§  21),  lay  off  from  it  the  proper  tangent  deflection  Lf 
for  the  given  sub-chord,  making  DF  of  the  given  length,  and  F  will 
be  a  point  on  the  curve.     The  proper  tangent  deflection  for  the  sub- 
chord  may  be  found  thus.     Eepresent  the  sub-chord  by  c',  and  the  cor- 

responding  chord  deflection  by  d',  and  we  have  (§  18)  5  c/'  =  —  ;  but 
since  hd  =  — '  we  have  ^  c/' :  2  cZ  =  c'- :  c^.     Therefore  ^d'  =  ,^d(-] 

Example.  Given  the  intersection  angle  I  between  two  tangents 
equal  to  16°  30',  and  R  =  12.')0,  to  find  T,  c/,  and  the  length  of  the 
curve  in  stations.     Here 

(§4)     T=R  tan.  j^  /=  1250  tan.  8°  15'  =  181.24  ; 

c'i         100*2 


*  Tlie  distance  B  M  is  not  exactly  equal  to  the  chord,  but  the  error  arising  from 
taking  it  equal  is  too  .small  to  be  regarded  in  any  curves  but  those  of  very  small 
radius.    If  necessary,  the  true  length  of  B  M  may  be  calculated ;    for  B  M  =: 


ORDINATES.  ,  11 

0  9)     sin.  L>  =  -f  ==  -|?-  =  .04  .=  nat.  sin.  2°  17^'; 

,  r  ,  ox  M         8 '  15'  495' 

(6  13)   ;z  =  —  = = =  3.60. 

^^       ^  n         2J17J'        137.5' 

These  results  show,  that  the  tangent  point  A  (fig.  3)  on  the  first  taii 
gent  is  18124  feet  from  the  point  of  intersection,  —  that  tlie  tan<.en\ 
deflection  G B=^ld=  A  feet,  —  that  the  chord  deflection  //Cor  K D 
=  8  feet,  —  and  that  the  curve  is  360  feet  long.  The  three  whole  sta- 
tions B^  C.  and  D  having  been  found,  and  the  tangent  D  L  drawn,  the 
tangent  deflection  for  the  sub-chord  of  60  feet  will  be,  as  shown  above, 

h  cV  =  4  C"-  )  =  4  X  .62  =  4  X  .36  =  1  44.     LF=  1.44  feet  being 

laid  off  from  DL,  the  point  F  will,  if  the  work  is  correct,  fall  upon 
the  second  tangent  point.  A  tangent  at  F  may  be  found  (§  21)  by 
producing  DF  to  P,  making  FP=  DF=  60  feet,  and  laying  ofl 
PN  =  1.44  feet.  FN  will  be  the  direction  of  the  required  tangent, 
which  should,  of  course,  coincide  with  the  given  tangent. 

23.  CurA^es  may  be  laid  out  with  accuracy  by  tangent  and  ch.ord 
deflections,  if  an  instrument  is  used  in  producing  the  lines.  But  if  an 
instrument  is  not  at  hand,  and  accuracy  is  not  important,  the  lines  may 
be  produced  by  the  eye  alone.  The  radius  of  a  curve  to  unite  two 
given  straight  lines  may  also  be  found  without  an  instrument  by  §  73, 
or,  having  assumed  a  radius,  the  tangent  points  may  be  found  by  §  74. 

C     Ordinates. 

24.  The  preceding  methods  of  laying  out  curves  determine  points 
100  feet  distant  from  each  other.  These  points  are  usually  sufficient 
for  grading  a  road  ;  but  when  the  track  is  laid,  it  is  desirable  to  have 
intermediate  points  on  the  curve  accurately  determined.  For  this  pur- 
pose the  chord  of  100  feet  is  divided  into  a  certain  number  of  equal 
parts,  and  the  perpendicular  distances  from  the  points  of  division  to 
the  curve  are  calculated.  These  distances  are  called  ordinates.  If  the 
chord  is  divided  into  eight  equal  parts,  we  shall  have  points  on  the 
curve  at  every  12.5  feet,  and  this  will  be  often  enough,  if  the  rails, 
which  are  seldom  shorter  than  15  feet,  have  been  properly  curved 
(§  28). 

25.  Problem.  Given  the  dpflection  angle  D  or  the  radius  R  of  a 
came,  to  Jind  the  ordinates  for  any  chord. 

Solution.  I.  To  find  the  middle  ordinate.  Let  AEB  (fig.  4)  be 
ft  portion  of  a  curve,  subtended  by  a  chord  A  B,  which  may  be  de- 


i'^ 


CIRCULAR    CURVES. 


noted  by  c.     Draw  the  middle  ordinate  ED,  and  denote  it  by  m.    Pro- 
duce ED  to  the  centre  F,  and  join  A F  and  A E.    Then  (Tab.  X.  3'« 


I Xu 


ED 

Id 


=  tan.  E  A  D,  or  E  D 


But,  since  the  angle 


E AD  is  measured  by  half  the  arc  BE,  or  by  half  the  equal  arc  AE^ 
we  have  EAD=hA FE.     Tlierefore  E  D  =  AD  tan.  ^  A FE,  ox 

^  m^  hciVin-^AFE. 

When  c  =  100,  A  FE  =  /)  (§  "),  and  m  =  50  tan.  5  /),  whence  7/) 
may  be  obtained  from  the  tabic  of  natural  tangents,  by  <liA-iding  tan 
4  Z)  by  2,  and  removing  the  decimal  point  two  places  to  the  right. 

The  value  of  m  may  be  obtained  in  another  form  thus.  In  the 
triangle  ADF we  have  DF=  ^A  F^  —  A  if-  =  ^72^  _  ^ ^2.  Then 
m  =  EF—  DF=  R  —  DF,  or 


7)1 


=  R  —  s/R- 


4  ^   • 


II.    To  find  any  other  ordinate,  as  i?iV,  at  a  distance  DN  =h  from 
ihe   centre  of  the  chord.     Produce  RN  until  it  meets  the  diameter 


parallel  to  ^  ^  in  G,  and  join  R  F.  Then  RG=  ^R  F^  —  F  G*  = 
y^-ZTp;  andRN=  RG  —  XG=  RG—  DF.  Substituting  the 
value  o?  RG  and  that  of  D  F  found  above,  we  have 


RN  =  ^R^  —  V'  -  ^R^  —  i  c2. 


ORDmATES.  13 

By  these  fcrmulaj  the  ordinates  in  Table  I  are  calculated. 

The  other  ordinates  may  also  be  found  from  the  middle  ordinate  by 
jie  following  shorter,  but  not  strictly  exact  method.  It  is  founded  on 
the  supposition,  that,  if  the  half-chord  B  D  he  divided  into  any  number 
of  equal  parts,  the  ordinates  at  these  points  will  divide  the  arc  E  B  into 
the  same  number  of  equal  j)arts,  and  upon  the  further  supposition,  that 
the  tangents  of  small  angles  are  proportional  to  the  angles  themselves. 
These  suppositions  give  rise  to  no  material  error  in  finding  the  ordi- 
nates of  railroad  curves  for  chords  not  exceeding  100  feet.  Making, 
for  example,  four  divisions  of  the  chord  on  each  side  of  the  centre,  and 
joining  A  B,  AS,  und  A  T,  we  have  the  angle  RAN=^EAD, 
since  R  B  is  considered  equal  to  %  E  B.  But  EAD=  iAFE. 
Therefore,  B.  A  N=  |  .1  FE.  In  the  same  way  we  should  find  SAO 
-=  ^  A  FE,  and  TA  P  =  ^  A  FE.  We  have  then  for  the  ordinates, 
R  N=^  AN  tan.  RAN  =  ^c  tan.  |  A  FE,  SO=AO  tan.  SA  0  = 
I  c  tan.  i  A  FE,  and  TP  =  AP  tan.  TAP  ^Ic  tan.  J  A  FE. 
But,  by  the  second  supposition,  tan.  %AFE  =  |  tan.  ^  AFE, 
tan.  \AFE  =  ^  tan.  i  A  FE,  and  tan.  ^AFE^\  tan.  i  A  FE. 
Substituting  these  values,  and  recollecting  that  §■  c  tan.  ^  A  FE  =  m, 
rte  have 

f    72iV=  |g  X  i  c  tan.  I  ^  jP^  =  jgwi, 

SO^\x^ctaxi.^AFE  =  \  vi, 

7  7 

TP  =  jg  X  i  c  tan.  ^  A  FE  =  ^g  m. 

In  general,  if  the  number  of  divisions  of  the  chord  on  each  side  of 
the  centre  is  represented  by  n,  we  should  find  for  the  respective  ordi- 

.                                             (n  +  l)(n-l)m     (n+2)(7t-2)m 
nates,  begmnmg  nearest  the  centre,  ■ — ■ :^ ,  ^^^ ? 

;n  +  3){n  — 3)wz 


«2 


,   &C. 


Example     Find  the  ordinates  of  an  8°  curve  to  a  chord  of  100  feet. 

Here  wi  =  50  tan.  2°  =  1.746,  TZiV^  ^  w  =  1.637,  6' 0  =  \m  ^  1..310, 

7 
and  TP  =  ^  w  =  0.764. 

26.  An  approximate  value  of  m  also  may  be  obtained  from  the  for- 
mula m  =  R  —  ^R^  —  \  c^     This  is  done  by  adding  to  the  quantity 

under  the  radical  the  very  small  fraction  g,  ^j  •>  making  it  a  perfect 


CIRCULAR    CURVES. 


f  quare,  the  root  of  which  will  he  R  —  5-5 .    Wc  have,  then,  n.  «=>  fi 


i^-n-J- 


8R 
SB.) 


8  R 


27.  From  this  value  of  m  we  see  that  the  middle  ordinates  of  any 
two  chords  in  the  same  curve  are  to  each  other  nearly  as  the  squares 
of  the  chords.  If,  then,  A  E  (fig.  4)  be  considered  equal  to  ^  y4  S.  its 
middle  ordinate  C //  ==  {ED.  Intermediate  points  on  a  curve  m;iy, 
therefore,  be  very  readily  obtained,  and  generally  with  sufficient  accu- 
racy, in  the  following  manner.  Stretch  a  cord  from  A  to  B,  and  Ijy 
means  of  the  middle  ordinate  determine  the  point  E.  Then  stretch 
the  cord  from  A  to  E,  and  lay  off  the  middle  ordinate  C 11  =  \  ED, 
thus  determining  the  point  C,  and  so  continue  to  lay  off  from  the  .'^■i;-- 
ressive  half-chords  one  fourth  the  preceding  ordinate,  until  a  sufficicru 
number  of  points  is  obtained. 

D.     Curving  Rails. 

28.  The  rails  of  a  curve  are  usually  curved  before  they  are  V.vkx  To 
do  this  properly,  it  is  necessary  to  know  the  middle  ordinate  of  the 
curve  for  a  chord  of  the  lenjith  of  a  rail. 

29.  Problem.  Given  the  radius  or  deflection  angle  of  a  curve.,  to 
find  the  middle  ordinate  for  curving  a  rail  of  given  length. 

Solution.    Denote  the  length  of  the  rail  by  Z,  and  we  have  (§  25) 

the  exact  formula  m  =  R  —  ^/FC^  —  4  ^'>  and  (§  26)  the  approximate 
formula 

m  —  ^ 


2R 


This  formula  is  always  near  enough  for  chords  of  the  lengtli  of  a  rail 

50 
If  we  substitute  for  R  its  value  (§  10)  R  —  sin^  '  ^^®  have, 

100 


Example.    In  a  1°  curve  find  the  ordinate  for  a  rail  of  18  feet  m 
length.     Here  R  is  found  by  Table  I.  to  be  5729.6.5,  and  therefore. 


KliVERSED    AND    COMPOUND    CURVES. 


13 


9- 
by  the  first  foi-mula,  m  --  11459.3  =  .00707.      By  the  sccorul   forniula, 

m  =  .81  sin.  30'  =  .00707.     The  exact  formula  would  give  the  same 
result  even  to  the  fifth  decimal. 

By  keeping  in  mind,  that  the  ordinate  for  a  rail  of  18  feet  in  a  1=^ 
curve  is  .007,  the  corresponding  ordinate  in  a  curve  of  any  other  de- 
gree may  be  found  with  suflficient  accuracy,  by  multiplyiug  tliis  deci- 
mal by  the  number  expres.sing  tlio  degree  of  the  curve.  Thus,  for  a 
curve  of  5'^  36'  or  5.6°,  the  ordinate  would  be  .M7  X  •'>-6  =  .0."9  ft.  =- 

468  in. 

For  a  rail  of  20  feet  we  have  ^  /^  =  100,  and,  consequently,  ?h  =- 
sin.  D.  This  gives  for  a  1°  curve,  m  =  .0087.  The  corresponding  or- 
dinate in  a  curve  of  any  other  degree  may  be  found  with  sufficient 
accuracy,  l^y  multiplying  this  decimal  by  the  number  expressing  the 
degree  of  the  curve. 

By  the  above  formula  for  m,  the  ordinates  for  curving  rails  in  Table 
I,  are  calculated. 


Article  II.  —  Reversed  and  Compound  Curves. 

30.  Two  curves  often  succeed  each  other  having  a  common  tangeni 
at  the  point  of  junction.  If  the  curves  lie  on  opposite  sides  of  the  com- 
mon tangent,  they  form  a  reversed  curve,  and  their  radii  may  be  the 
!,ame  or  different.    If  they  lie  on  the  same  side  of  the  common  tangcTit 


tney  have  different  radii,  and  form  a  compound  curve.     Thus  A  B  C 
'fiff.  .5")  is  a  reversed  cirve,  and  .1  B  D  %  comoound  curve. 


16 


CIRCULAR    CURVES. 


31,  ProbleiJl.      To  lay  out  a  reversed  or  a  compound  cun>e,  tufien. 
the  radii  or  dejiection  anyles  and  the  tangent  points  are  known. 

Solution.  I/ay  out  the  first  portion  of  the  curve  from  A  to  B  Cfig.  5), 
by  one  of  the  usual  methods.  Find  B  F,  the  tangent  to  A  B  at  the 
point  B  (§  16  or  ^  21).  Then  B  F  will  be  tlie  tangent  also  of  the  sec- 
ond portion  B  C  oi  a  reversed,  or  Zi  D  of  a  compound  curve,  and  from 
this  tangent  cither  of  these  portions  may  be  laid  ofl'  in  the  usual  man 
ner 

A.     Reversed  Curves. 

32    I'SieOJ'CRi.      Tlie   reversing  point  of  a  reversed  curve  letwces 
parallel  tangents  is  in  the  line  joining  the  tangent  points. 


Fig.  6. 


t\ 


Demonstration.  Let  A  CB  (fig.  6)  be  a  reversed  curve,  uniting  tin 
parallel  tangents  HA  and  B  K,  having  its  radii  equal  or  unequal,  and 
reversing  at  C.  If  now  the  chords  A  Cam]  CB  are  drawn,  we  have 
to  prove  that  these  chords  are  in  the  same  straight  line.  The  radii 
E  C  and  C  F,  being  perpendicular  to  the  common  tangent  at  C  (§  2, 1.), 
are  in  the  same  straight  line,  and  the  radii  A  E  and  B  F,  being  per- 
pendicular to  the  parallel  tangents  HA  and  B  K,  are  parallel.  There- 
fore, the  angle  AE  C=  CFB,  and,  consequently,  E  C A,  the  half 
supplement  of  A  E  C,  is  equal  to  F  C  B,  the  half  supplement  of  CFB; 
but  these  angles  cannot  be  equal,  unless  A  Cand  C  B  are  in  the  same 
straight  line. 

33.  Proljlem.  Given  the  perpendicular  distance  between  two  par- 
alM  tangents  B  D  =^  b  {Jig  6),  and  the  distance  between  the  two  tangeni 
voints  A  B  =  a,  to  determine  the  reversing  point  C  and  the  common  radnti 
E  C  ^  C  F  =  R  of  a  reversed  curve  uniting  the  tangents  HA  and  B  K. 

Solution.     Let   ACB  be  the  required  curve.     Since  the  radii  are 


REVERSED    CURVES. 


n 


equal,  and  the  angle  AE  C  =  B F C,  the  triangles  AE  C and  B  FC 
are  equal,  and  A  C  =  CB  ^  ^a.  The  reversing  point  C  is,  therefore, 
the  middle  point  of  A  B. 

To  find  R,  draw  E  G  perpendicular  to  A  C.  Then  the  right  tri- 
itngles  AEG  and  BAD  are  similar,  since  (§  2,  III.)  the  angle 
BAD  =  hAEC^  AEG.  Therefore  A  &  -.  A  G  ^.  AB  :  BD, 
or  ii  :  ^  a  =  a  :  6  ; 

46 

Corollary.  If  R  and  h  are  given,  to  find  a,  the  equation  72  =  j^ 
gives  a'  =  4  Rb; 


a 


2  JR  b. 


Examples.      Given  6  =  12,  and  a  =^  200,  to  determine  R.     Here 
2002         10000 

12     ~  833|. 


/i'  = 


4X12 


Given  R  =  675,  and  b  =  12,  to  find  a.  Here  a  =  2^675  X  12  = 
2y8T00  ==  2  X  90  =  180. 

34.  Protolem.  Given  the  perpendicular  distance  between  two  par- 
allel tangents  B  D  =  b  {fig-  7),  the  distance  between  the  two  tangent  points 
A  B  =  a,  and  the  first  radius  E  C  =  R  of  a  reversed  curve  uniting  the 
tangents  HA  and  B  K.  to  find  the  chords  A  C  ~  a'  and  C  B  =  a",  and 
the  second  radius  CF  =  R'. 


Solution.    Draw  the  perpendiculars  E  G-  and  FL.    Then  the  right 
triangles  A  B  D  and  E  A  G  are   similar,  since  the  angle  B  AD  ^ 


i8  CIRCULAR    CURVES. 

iAEC=  AE  G.    Therefore  AB  :  B D  =  E  A  :  A  G,  or  a  :  b 

2Rb 


a 

Since  a'  and  a"  are  (§  32)  parts  of  a,  we  have 

a"  =  a  —  a'. 


To  find  R'  the  similar  triangles  A  B  D  and  F  B  L  give  A  B  :  B  D 
=  F  B  :  B  L,ox  a  :b  =  R' '.  ^  a" ; 

a  a" 

Example.     Given  6  =  8,  a  =  160,  and  R  =  900,  to  find  a',  a",  and 

/?'.     Here  a'  = ^ =  90,  a"  =  160  —  90  =  70,  and   R<  = 

160  X  70        _^^ 
-2X8     =700. 

35.  Corollary  1.     If  6,  a',  and  a"  are  given,  to  find  a.  A',  and  A' , 
we  have  (§  34) 

^  a  =  a'  +  a"  ;        R=—;        R'  =  1^. 

2  6  26 

Example.    Given  6  =  8,  a'  =  90,  and  a"  =  70,  to  find  a,  A,  and  R 

Here  a  =  90  -f  70  =  160,  A  =  -g  j<  8    =  900,  and  A'  =     ^xS    = 
700. 


36.  Corollary  2.     If  A,  A',  and  h  are  given,  to  find  a,  a',  and  a'\ 

c  have  (§  35),  A  • 
B«=  26  (A +  ^'); 


V     r>     I     -ni        aai  +  aa"        a  (a' -fa'')         a2 
wc  have  (§  35),  R  -j-  R'  =        ^b —   ~  — 2b —  ~  2b-     Therefore 


.'.a  =  y2  6(A-t-A'). 
Having  found  a,  we  have  (§  34) 

a  a 

Example.     Given  A  =  900,  A'  =  700  and  6  =  8,  to  find  a,  a',  aim 

a".     Here  a  =  ^2  X  8  (900  +  700)  =  ^16  X  1600  ^-  160,  a'  =- 

8  X  900  X  8        ^^         -     ,,        2X700X8        _ 
— jg^j =  90,  and  a"  = ^ =  70. 


REVERSED    CURVES. 


19 


37.  Problem.  Given  the  angle  A  K  B  =  K,  which  shows  the 
change  of  direction  of  two  tangents  HA  and  B  K  {fig.  8),  to  unitr.  these 
tangents  by  a  reversed  curve  of  given  common  radius  R,  starting  from  a  giv- 
en tangent  point  A. 


3l 

B  K 

^^  Fig.  8. 

Solution.  With  the  given  radius  run  the  curve  to  the  point  Z),  where  the 
tangent  D  N becomes  parallel  to  D  K  The  point  D  is  found  thus.  Since 
the  angle  N  G  K,  which  is  double  the  angle  II A  D  {(j  2,  II.),  is  to  be 
made  equal  to  A  KB  =  K,  lay  off  from  FIA  the  angle  HA  D=\E 
Measure  in  the  direction  thus  found  the  chord  AD  =  2R  sin.  ^  75: 
This  will  be  shown  (§  69)  to  be  the  length  of  the  chord  for  a  deflection 
angle  ^  K.  Having  found  the  point  D,  measure  the  perpendicular  dis- 
tance D  M  =  b  between  the  parallel  tangents.    ■ 

The  distance  DB  =  2DC  =  a  may  then  be  obtained  from  the  for- 
muln  r§  3.3,  Cor.) 

l^  a  =  2  ^ITb  . 

The  second  tangent  point  B  and  the  reversing  point  Care  now  ue- 
tcrniined.  The  direction  o(  D  B  or  the  angle  B  DNmnj  also  I)e  ob- 
tained ;  for  sin  BDN 


sin,  DBM  =  TTiF,,  or 


sin.  BDN 


b 

a 


38.  Problem.  Given  the  line  A  B  =  a  {fig.  9)  which  Joins  the 
fixed  tangent  points  A  and  B,  the  angles  HAB  =  A  and  ABL  =  B, 
tnd  the  first  radius  A  E  =  R,  to  find  the  second  radius  B  F  =  R  of  a 
Teversfd  curve  to  unite  the  tangents  H'  A  and  B  K. 

First  Solution.  With  the  given  radius  run  the  carve  to  the  point  Z), 
ohere  the  tangent  D  N  becomes  parallel  to  B  K.     The  point  D  is  found 


20 


CIRCULAR    CURVES. 


thus.  Since  the  angle  H  G  N,  which  is  double  HA  D  (§  2,  II.),  is 
equiil  to  J.  CO  S,  lay  off  from  HA  the  angle  HA  D  —  ^  (At^  B),  and 
measure  in  this  direction  the  chord  A  D  =  2  R  sin.  ^  (A&o^)  (§  69) 


Setting  the  instrument  at  Z),  run  the  curve  to  the  reversing  point  C  in  the 
line  from  D  to  B  {^  32),  and  measure  D  C  and  C  B.  Then  the  similar 
triangles  DEC  and  BFC  give  DC:DE  =^  CB  :  B  F,  or  D  C :  Ji 
^  CB:R'; 

CB 


.R'^. 


DC 


X  R- 


Second  Solution.  By  this  method  the  second  radius  may  bt  founu 
by  calculation  alone.  The  figure  being  drawn  as  above,  we  have,  in 
the  triangle  A  B D,  AB  =  a,  AD  =  2R  sin.  ^  (A  —  B),  and  the 
included  angle  DAB  =  HA  li  ~  HAD  =  A  —  h  (A  —  B)  ^ 
^  {A  -\-  B).  Find  in  this  triangle  (Tab.  X.  14  and  \2)  B D  and  the 
angle  ABD.     Find  also  the  angle  DBL^B-\-ABD. 

Then  the  chord    C B  =  2  R'  sin.  hBFC  =2R'  sin.  D B  L,  and 


the  chord  D  G 
CB  =  BD  — 
DBL, 


=  2R  sin.  ^DE  C  =  2R  sin.  DBL  (§  69).      But 
D  C;  whence  2  R'  sin.  D  B  L  =  B  D  —  2  R  sm 


.R> 


BD 


2  sin.  DBL 


—  R. 


"When  the  point  D  falls  on  the  other  side  of  A,  that  is,  when  the 
angle  B  is  greater  than  jl,  the  solution  is  the  same,  except  that  the 
mgle  DAB  is  then  180°  —  ^(A  -\-  B),  and  the  angle  DBL=  B  — 
ABD. 


REVERSED    CURVES. 


21 


39.  Probloiia.  Given  the  length  of  the  common  tangent  D  G  —  a^ 
and  the  angles  of  intersection  I  and  I'  (Jig.  10),  to  determine  the  common 
radms  C  E  =  C  F  =  li  of  a  reversed  curve  to  urate  the  tangents  II A 
rtnn  B  L. 


Fig.  10. 


T- 


Solution.    By  §  4  wc  have  DC  =  R  tan.  |  /,  and  CG=  R tan.  |  /' . 
whence  R  (tan.  ^  /  +  tan.  hi')  =  D  C  -{-  C  G  =  a,  or 

R  = 


tan.  ^  /  +  ti^n-  k  ^' 
This  formula  may  be  adapted  to  calculation  by  logarithms ;   for  we 

have  (Tab.  X.  35)  tan.  ^7+  tm.^P  =  ^T.'^^jcot  fj-    Substituting 
this  value,  we  get 

rw  R  -  «gos.  ^7cos.  ^7^ 

sin.i(^+/0 

The  tangent  points  A  and  B  are  obtained  by  measuring  from  D  a 
iistance  J.  Z)  =  72  tan  |-  7,  and  from  G  a  distance  B  G  —  R  tan.  \  I', 

Example.     Given  a  =  600,  1  =  12°,  and  F  =  8°  to  find  R.     Here 

a  =  600  2.778151 

i7=6°  cos.  9.997614 

f  7'  =  4°  cos.  9.998941 


R  =  3427.96 


2.774706 
sin.  9.239670 


3.535036 


22 


CIRCULAR    CURVES 


40.  Problem.  Given  the  line  AB  =  a  {fig-  10),  which  jchis  the 
fixed  tangent  points  A  and  B,  the  angle  DAB  =  A,  and  thr  angle 
A  B  G  =  B,io  Jind  the  common  radius  E  C  =  CF  =  Rof  ar,  versed 
■:urre  to  unite  the  tangents  HA  and  B  L. 


Solution.  Find  Jirst  the  auxiliai-y  angle  A  K E  =  B  KF,  ivhich  inmj 
be  denoted  by  K.  For  this  purpose  the  triangle  A  E  K  gives  A  E:  E  K 
=  sin.  K :  sin.  E  A  AT.  Therefore  E  K  sin.  K  =  A  E  sin.  E  A  K  ~ 
R  cos.  A,  since  EAK  =  90^  —  A.  In  like  manner,  the  triangle 
BFK  gives  FK  sin  K=  BF  sin.  FBK  =  R  cos.  B.  Adding 
these  equations,  we  have  {E K-\-  FK)  sin.  K=  R  (cos.  J.  -\-  cos.  B), 
or,  since  E  K  +  FK  =  2  R,  2  R  sin.  K  =  R  (cos.  A  +  cos.  B) 
Therefore,  sin.  K  =  ^  (cos.  A  -\-  cos.  B).  For  calculation  by  loga- 
rithms, this  becomes  (Tab.  X.  28) 

sin  K  =  cos.  i(A-\-  B)  cos.  ^(A  —  B). 

Having  found  K,  we  have  the  angle  AE K  =  E  =  18(P  —  K  — 
EAK=  \%QP  —  K  —  (90^  —  yl)  =  90°  +  ^  —  Z;  and  the  angle 
BFK=  F=  180°— K— FBK  =  180°  —  TT— (90=— J5)  =  90- 
-{-  B  —  K  Moreover,  the  triangle  A  E  K  gives  Ah  A  K  = 
sin.  K:  sin.  E,or  R  sin.  E=  J..K'sin.  K  and  the  triangle  B  F K  gives 
BF:BK  =  s\n.K:  s'm.F,  or  R  sin.  F  =  B K  sin.  K.  Adding  these 
equations,  we  have  R  (sin.  E  -f-  sin.  F)  =  (A  K  -j-  B  K)  sin.  K  — 
a  sin.  K.     Substituting  for  sin.  E  4-  sin.  Fits  value  2  sin.  ^  {E  -j-  l^ 


COMPOUND    CURVES.  2S 

(^g_  ^  (E  —  F)  (Tab.  X.  26),  we  have  2  li  sin.  .^  (A'  -|-  F)  cos. 

i  a  sin.  K  „. 

^(£'_F)=asin.A:.    Therefore R  =  ^1^77(5-4.  f)^os.  U^- -F) '  *'* 

nally,  substituting  for  A'  its  value  90°  -f  ^  —  A',  and  for  Fits  value 

grjo  Lj.  B  —  A',  we  get  h  {E  +  F)  =  90°  —  [A'  —  ^  (.1  +  13)1  an  J 
,^  (A'  —  F)  =  i  (yl  —  /}) ;  whence 


COS.  [K-  ^  {A  4-  Z^)J  cos  ^(A  —  B) 

Eximple.     Given  a  =1500,  A  =  18°,  and  B  =  6°,  to  find  /.'.     Here 
^  (^  4-  C)  =  12°  cos.  9.990404 

i  (^1  —  i?)  =  6°  cos  9  997614 

A'  =  76°  36'  10"  sin.  9.988018 

Aa  =  750  2.87.^)001 


K  -^{A-^-  B)  ^  64°  36'  10       COS.  9.632347 
J(^  — /?)  =  6*'  COS.  9.9976  U 


2.863079 


9.629961 


/{=  1710.48  3.233118 


B.     Compound  Curves. 

41.  irhcorem*  If  one  branch  of  a  compound  curve  he  produced^ 
HJilil  the  tangent  at  its  extremity  is  parallel  to  the  tangent  at  the  extremity 
of  the  second  branchy  the  common  tangent  point  of  the  tico  arcs  is  in  the 
straight  line  produced,  which  passes  through  the  tangent  points  of  these  par- 
allel tangents. 

Demonstration.  Let  A  CB  (fig.  11)  be  a  compound  curve,  uniting 
the  tangents  HA  and  B  K.  The  radii  C'F  and  C'F,  being  perpen- 
dicular to  the  common  tangent  at  C  (§  2, 1.),  are  in  the  same  straight 
line.  Continue  the  curve  A  C  to  Z>,  where  its  tangent  OD  becomes 
parallel  to  B  K,  and  consequently  the  radius  DE  parallel  to  B  F. 
Then  if  the  chords  CD  and  CB  be  drawn,  we  have  the  angle  CE D 
=  CFB;  whence  E  CD,  the  half-supplement  of  C E  D,  is  equal  to 
F CB,  the  half-supplement  of  CFB.  But  E  CD  cannot  be  equal  to 
F  C B,  unless  CD  coincides  Avith  CB.  Therefore  the  line  B  D  pro- 
inced  passes  through  the  common  tangent  point  C 


24 

42.  Problem. 

compound  curve. 


CIRCULAR    CURVES. 


To  find  a  limit  in  one  direction  of  each  radius  of  ol 


!S)lution.  Let  A I  and  Bl  (fig.  11)  be  the  tangents  of  the  curve. 
Through  the  intersection  point  7,  draw  IM  bisecting  the  angle  A  IB. 
Draw  A  L  and  B  M  perpendicular  respectively  to  A  I  and  B  /,  niect- 
ing  1 M  in  L  and  M.  Then  the  radius  of  the  branch  commencing  on 
the  shorter  tangent  A  /must  be  less  than  AL^  and  the  radius  of  the 
branch  commencing  on  the  longer  tangent  B  I  must  be  greater  than 
BM.  For  suppose  the  shorter  radius  to  be  made  equal  to  A  L^  and 
make  IN  =  A  I,  and  join  L  N.  Then  the  equal  triangles  A  IL  and 
NIL  give  A  L  =  L  N;  so  that  the  curve,  if  continued,  will  pass 
through  iV,  where  its  tangent  will  coincide  with  IN.  Then  (§  41)  the 
common  tangent  point  would  be  the  intersection  of  the  straight  line 
through  B and  iVwith  the  first  curve;  but  in  this  case  there  can  be  no 
intersection,  and  therefore  no  common  tangent  point.  Suppose  next, 
that  this  radius  is  greater  than  A  L,  and  continue  the  curve,  until  its 
tengent  becomes  parallel  to  BI.    In  this  case  the  extremity  of  the 


COMPOUND    CURVES.  25 

curve  will  fall  outside  the  tangent  BIm  the  line  A  iV produced,  and  a 
straight  line  through  D  and  this  extremity  will  again  fail  to  intersect 
the  curve  already  drawn.  As  no  common  tangent  point  can  be  found 
when  this  radius  is  taken  equal  to  A  L  or  greater  than  A  L,  no  com- 
pound curve  is  possible.  This  radius  must,  therefore,  be  less  than  A  L. 
In  a  similar  manner  it  might  be  shown,  that  the  radius  of  the  other 
branch  of  the  curve  must  be  greater  than  B  M.  If  we  suppose  the  tan- 
gents A  I  and  B  J  and  the  intersection  angle  /  to  be  known,  we  have 
{^  5)  A  L  =  A  I  cot.  ^  /,  and  B  M  =  B  I  cot.  ^  7.  These  values  are 
therefore,  the  limits  of  the  radii  in  one  direction. 

43.  If  nothing  were  given  but  the  position  of  the  tangents  and  the 
tangent  points,  it  is  evident  that  an  indefinite  number  of  different  com- 
pound curves  might  connect  the  tangent  points ;  for  the  shorter  radius 
might  be  taken  of  any  length  less  than  the  limit  found  above,  and  a 
corresponding  value  for  the  greater  could  be  found.  Some  other  con- 
dition must,  therefore,  be  introduced,  as  is  done  in  the  following 
problems. 

44.  Problem.  Given  the  line  AB  =  a  {Jig.  11),  which  joins  Oie 
fixed  tangent  points  A  and  B,  the  angle  B  A  I  =^  A,  the  angle  AB I  = 
D,  and  the  first  radius  A  E  —  B,  to  find  the  second  radius  B  F  =  R'  of 
a  compound  curve  to  unite  the  tangents  HA  and  B  K. 

Solution.  Suppose  the  first  curve  to  be  run  with  the  given  radius 
from  A  to  Z),  where  its  tangent  DO  becomes  parallel  to  BI^  and 
the  angle  IAD  =  i  (^  -f-  B).  Then  (§  41)  the  common  tangent 
point  C  is  in  the  line  B  D  produced,  and  the  chord  CB  =  CD  -j- 
B  D.  Now  in  the  triangle  AB  D  we  liave  A  B  ^  a^  AD  =  2  R 
sin.  ^  {A  -\-  B)  (§  69),  and  the  included  angle  D  A  B  =^  I A  B  — 
IAD  =  A  —  ^  (A  -{-  B)  ^  ^  {A  —  B).  Find  in  this  triangle 
(Tab.  X.  14  and  12)  the  angle  A  B  D  and  the  side  B  D.  Find  also  the 
angle  CBI=B  —  ABD. 

Then  (^  69)  the  chord  CB  ^  2  R'  sin.  CB  Z,  and  the  chord  CD  = 
2  R  sin.  CD0=2R  sin.  CBI.  Substituting  these  values  of  CB 
and  CD  in  the  equation  found  above,  C B  =  CD  -\-  B  D,  we  have 
2/^'  sin.  CZ3/=  2  R  sin.  CBI+BD; 

l^  -.R'  =  R+        ^^ 


2  sin.  CBI 


When  the  angle  B  is  greater  than  A,  that  is,  when  the  greater  radius 
»8  given,  the  solution  is  the  same,  except  that  the  angle  D  A  B  ^ 


26  CIRCULAR    CURVES. 

J  (D  —  A),  and  C BI'is  found  by  snbtracting  the  supplement  oi  A  Li  IJ 
from  B.    We  shall  also  find  CB  —  CD  —  B  D^  and  conscauenilv 

^'^    ~  ^^        2  sin.  CBI' 

If  more  convenient,  the  point  D  may  be  determined  in  the  field,  by 
laying  otf  the  angle  I A  D  =  ^  [A  -j-  B),  and  measuring  the  distance 
A  D  -^  2  R  sin.  i  ( J.  -j-  B).  BD  and  CB  I  may  then  be  measured, 
instead  uf  being  calculated  as  above. 

Example.  Given  a  =  950,  ^  =  S^',  /5  =  7^,  and  R  =  3000,  to  find 
A''.  Here  AD  =  2  X  3000  sin.  h  (S^  +  7°)  =  783.16,  and  DA  B  -= 
^  (8°  —  7°)  =  30'.     Then  to  find  .1  B  D  we  have 

AB  —  A  D  ^  166.84  2.222300 

i  (J  DB  -\-  A  B  D)  =  89°  45'  tan.  2.360180 

4.582480 
A  B  -j-  AD  =  1733.16  3.238831 

I  (.1  Z)  L.'  —  .1  Z;  D)  =  87°  24'  17"    tan.  1.343641 

.'.ABD  =  2°  20' 43" 
Next,  to  find  B  D, 


A  D  ==  783.16* 

2.893849 

DAB  =  30' 

sin 

7.940845> 

0.834691 

ABD  =  2°  20'  43" 

sin 

8.611948 

jBZ)  =  167.01 

2.222743 

B- 

-ABD=  CBI  =  4^S9'  17" 
2  (/2'  —  R)  ^  2058.03 

sin. 

8.909292 

3.313451 

.-.R'  —  R  =  1029.01 

R'  ^ 

=  3000  4-  1029.01  =-  4029.01 

To  find  the  central  angle  of  each  branch,  we  have  CI  B  —  2  C  B 1 
=  9°  18' 34",  which  is  the  central  angle  of  the  second  branch;  and 
AEC=AED-CED  =  Ai-B  —  2CBl  =  5°  41' 26",  which 
is  the  central  angle  of  the  first  branch 

45.  Problem.     Given  (Jig.  11)  the  tangents  Al=  T,  B I  =  T', 

the  angle  of  intersection  =  /,  and  the  first  radius  A  E  =  R,  to  find  the 
second  radius  B  F  =  R'- 

Solution.  Suppose  the  first  curve  to  be  run  with  the  given  radius 
from  A  to  D,  where  its  tangent  D  0  becomes  parallel  to  Bl.    Through 


COMPOUND    CURVKS.  27 

D  draw  D  P  parallel  to  .4  7,  and  v/c  have  IP  =  DO  =  AO  = 
R  tan.  ^  7  (M)-  Then  in  the  triangle  D  P  B  vre  have  D  P  =  1 0  =^ 
A1—A0=  T  -  R  tan.  ^I,  BP=BI— IP=T'— R  tan.  ^  7, 
and  the  included  angle  DPB  =  AIB  =  180'  —  7.  T'/nc/  m  //ijs  ^v- 
angle  the  angle  C  B  I,  and  the  side  B  D.  The  remainder  of  the  solution 
is  the  same  as  in  §  44.  The  determination  of  the  point  D  in  the  field 
is  also  the  same,  the  angle  IAD  being  hei'e  =  ^  7.  When  BU 
gi  cater  than  A,  that  is,  when  the  greater  radius  is  given,  the  solution  is 
d'.c  same,  except  that  D  P  =  R  tan.  ^  I  —  T,  and  B  P  =  R  tan.  1 1 

Example.  Given  T=  447  32,  T'  =  510. 84,  7  =  15°  and  R  =  3000, 
to  find  R'.  Here  7^1  tan.  |  7=  3000  tan.  7^°  =  394.96,  DP  =  447.32 
—  394.96  =  52.36,  BP  =  510.84  —  394.96  =  115.88,  and  DP  B  = 
180^  —  15°  =  165°.     Then  (Tab.  X.  14  and  12) 

SP  —  7)P  =  63.52  1.802910 

|(SDP  +  P^Z))  =  7=30'  tan.  9.119429 

0.922339 
.BP  +  7)P  =  16824  2.225929 

^{BDP  —  PBD)  =  2°  50' 44"    tan.  8696410 

.'.PBD=  C23 7  =  4"^  39' 16" 
Next,  to  find  B  7), 

Z)P=  52.36  1.719000 

DP  B  =  lb°  sin.  9.412996 


1.131996 
P  Z^  D  =  4°  39'  16"     sin.  8.909266 


2^7)  =167.005  2.222730 

Ibe  tangents  in  this  example  were  calculated  from  the  example  in 
^  44.  The  values  of  CB I  and  B  D  here  found  differ  slightlv  from 
those  obtained  before.  In  general,  the  triangle  DBP  is  of  better 
form  for  accurate  calculation  than  the  triangle  AD B. 

46.  If  no  circumstance  determines  either  of  the  radii,  the  condition 
may  be  introduced,  that  the  common  tangent  shall  be  parallel  to  the 
line  joining  the  tangent  points. 

Problem.  Given  the  line  AB  =  a  (Jig.  12),  which  unites  the 
fixed  tangent  points  A  and  B,  the  angle  I A  B  =  A,  and  the  angle 
A  B I  =  B,  to  find  the  radii  A  E  =  R  and  B  F  =  R'  of  a  compound 
'.urve^  having  the  common  tangent  I)  G  parallel  to  A  12 


28 


CIRCULAR    CURVES. 


Solution.    Let  A  C  and  B  C  be  the  two  brai^ches  of  the  required 
curve.  ar:d  draw  the  chords  A  C  and  B  C.    These  chords  bisect  the 


fig  12 


angles  A  and  B ;  for  the  angle  D  A  C  =  ^  ID  G  =  ^  I A  B,sluA  the 
angle  G  B  C  =--- ^  D  G 1  =  h  A  B  I.  Then  in  the  triangle  A  C B  wo, 
have  AC\AB^  sin.  A  BC  :  sin.  A  C B.  But  ACB=  180^  — 
(C^  Z5  +  CB  A)  =  180"  —  4  (^1  4-  B),  and  as  the  sine  of  the  sup- 
plement of  an  angle  is  the  same  as  the  sine  of  the  angle  itself. 
sin.  A  CB  =  sin.  \  {A -^  B).     Therefore  A  C :  a  =  sin.  ^  B  :  sin. 

^  (A  -f-  B),  or  A  C  =  sin"  1^4,  B)  •    ^^  ^  similar  manner  we  should 


a  sin.  ^  A 


iAC 


^^^  I^C=  ,in.  i  (A%  B) '    ^'o->^  ^^e  have  (§  68)  72  =  -^j^-p;  ,  and 
j^  ,  or,  substituting  the  values  of  ^  Cand  B  Cjust  found, 


U'  = 


jB  C 

sin. 


i  a  sin.  ;V  5 


a  sin. 


sin.  ^  A  sin.  ^  (^  +  5) '  sin.  ^  B  sin.  ^  ( A  +  i^) 


Example,     Given  a  =  950,  ^  =  8°,  and  B  =  7°,  to  find  R  and  /2' 
Here 


COMPOUND    CURVES'.  29 

i  a  =  475  2.676  '.94 

^  B  ^  3°  30'  sin.  8. 785675 


1.462369 
1^  =  4°  sin.  8.843585 

i  (A  +  ^)  =  7^  30'      sin.  9.115698 


7.959283 

R  =  3184.83  3.503086 

i  ransposing  these  same  logarithms  according  to  the  formula  for  R 
«e  hare 

?  «  =  475  1.676694 

•M  =  4°  sin.  8.843585 


h  B  =  3°  30'     sin.  8.785675 
^  (^  +  Z?)-=  7-=  30'     sin.  9  115698 


1.520279 


7.901373 


R'  =  4158.21  3.618906 

47.  Problem.  Glveji  the  line  AB  =  a  {Jig.  12),  wkich  unites  the 
fixed  tangent  points  A  and  B,  and  the  tangents  AI  =  T  and  BI  =  T', 
Mjind  the  tangents  AD  =  x  and  B  G  =  y  of  the  tico  branches  of  a  com- 
pound curve,  having  its  common  tangent  D  G  parallel  to  A  B. 

Solution.  Since  D  C  =  A  D  =  x,  and  C  G  =^  B  G  =  y,  we  have 
fjQ  =  x-j~jj.  Then  the  similar  triangles  IDG  and  lAB  give 
f  D  :  lA  =  D  G  :  A  B,  or  T  -  X  -  T  ■=  X  -\-  y  :  a.  Therefore 
aT  —  ax  =  Tx  +  Ty  (1).  Als(  ^  0  :  A  I  =  B  G  :  B  I,  or 
T:T  =  y:T'.  Therefore  Ty  =  T  r  (^').  Substituting  in  (1)  the 
ralue  of  Ty  in  (2),  we  have  a  T—  ax  ■.  7  r  +  2'' :r,  or  a  a:  +  Tor  -|- 
T'x  =  aT; 

a-\-T-\-T'' 

T'x 
and,  since  from  {2),y  =  -y-  , 


a-\-T-\-T' 


The  intersection  points  D  and  G  and  the  common  tangent  pomt  C 
are  now  easily  obtained  on  the  ground,  and  the  radii  may  be  found  by 
the  usual  methods.      Or,  if  the  angles   TAB  =  A  and  A  B I  -^  B 


30 


CIRCULAR    CURVES. 


have  been  measured  or  calculated,  we  have  (§  5)  R  =^  x  cot.  ^  A,  and 
R'  =  y  cot.  ^  B.     Substituting  the  values  of  x  and  y  found  above,  wa 

have  R  =  q^y^r' '  ^^"^  ^  =  M=^M=  T<  • 

Exampie.  Given  a  =  500,  T  =  250,  and  T'  =  290,  to  find  x  and  y 
Here  a  +  7  +  I'  =  500  +  250  +  290  =  lOtO  ;  whence  a:  =  500  >< 
250  -r  1040  =  120.19,  and  ^  =  500  X  290  -r-  1040  =  139.42. 

48.  Probleaea.  Given  the  tangents  Al  =  T,  Bl  =T',  and  tfu 
angle  of  intersection  /,  to  unite  the  tangent  points  A  and  B  (Jig.  13)  hy  a 
compound  curve,  on  condition  that  the  tivo  branches  shall  have  their  angles 
of  intersection  IDG  and  I  GD  equal. 


Fig   13 


^ututiirn.     feince  IDG  =  lGD  =  hl^yf&  have  I D  =  1  G.    Rep 
'■escnt  the  line  Ih  ~  1  Ghy  x.     Then  if  the  perpendicular  IHhe  let 

♦  The  radii  of  an  oval  of  given  length  and  breadth,  or  of  a  three-centre  arch  of  given 
epan  and  rise,  may  also  be  found  from  these  formulae     In  these  cases  A-^  B  =  90'-, 

and  the  values  of  R  and  R'  may  be  reduced  to  R  = — ; — ^^, 7;;^  and  R'  = 

aTi 


a+  T—  Ti 
calculated 


a+T'  —  T 
.    These  values  admit  of  an  easy  construction,  or  they  may  be  readily 


TURNOUTS    AND    CROSSINGS.  31 

fall  fiom  /,  we  !iave  (Tab.  X.  U)  D  H  =  I D  cos.  IDG  =  x  cos.  ^  i, 
sxiUDG^'lx  COS.  ^  /.  But  DG  =  DC^CG  =  AD-\-BG== 
7'  ~  2  +  T'  —  X  =  r  +  Ti  —  2x.  Therefore  2  a:  cos.  \l  = 
r  +   T'  —  2  .r,  or  2  T  +  2  .r  cos.  i  /  =-    T  -\-    T' ;    whence  jt  = 

j^:^^^;j^,or(lab.X.25) 


I^'  2r  = 


^{T+rO 


CO 


s.^i/ 


The  tangents  AD  =  T—  x  and  B  G  =-  T'  —  x  are  now  readilj 
(bund.  With  these  and  the  known  angles  of  intersection,  the  radii  oi 
deflection  angles  may  be  found  (§  5  or  §  11)  This  method  answers 
very  well,  when  the  given  tangents  are  nearly  equal ;  but  in  general 
•  he  preceding  method  is  preferable. 

Example.     Given  T  =r  480.  T'  ==  500,  and  7=18=,  to  find  :r.     Here 

^  (T  -\-  T')  =  2-L5  2.3891  G6 

^7=4=^  30'      2  COS.  9.997318 


X  =  246.52  2.391848 

Then  AD  =  480  —  246.52  =  233.48,  and  B  G  =-  500  —  246.52  -= 
253.48.  The  angle  of  intersection  for  both  branches  of  the  curve  being 
y°,  we  find  the  radii  AE  =  233.48  cot.  4^  30'  =  2P66.65,  and  B  F  ^= 
2'>3.t8  cot.  4°  30'  =  3220.77. 


Article  III. —  Turnouts  and  Crossixgs. 

49.  The  Uaual  mode  of  turning  off  from  a  main  track  is  by  switch- 
ing a  pair  of  rails  in  the  main  track,  and  putting  in  a  turnout  curve 
tangent  to  the  switched  rails,  wiih  a  frog  placed  where  the  outer  rail 
-^f  the  turnout  crosses  the  rail  of  the  main  track.  A  B  (fig.  14)  repre- 
sents one  of  the  rails  of  the  main  track  switched,  B  /''represents  the 
outer  rail  of  the  turnout  curve,  tangent  to  A  B,  and  E  shows  the  posi- 
lion  of  the  frog  The  switch  angle,  denoted  by  S,  is  the  angle  DAB, 
^urnled  by  the  switched  rail  A  B  with  A  D,  its  former  position  in  the 
main  track.  The  frog  angle,  denoted  by  E,  is  the  angle  G  EM  made 
Ijy  the  crossing  rails,  the  direction  of  the  turnout  rail  at  7^  being  the 
tangent  EM  at  that  jjoint.  In  the  problems  of  this  article  the  gauge 
ot  the  track  D  C.  denoted  by  g,  and  the  distance  D  B,  denoted  by  d 
are  supposed  to  be  known.  The  switch  angle  S  is  also  supposed  to 
bo  known,  since  its  sine  (Tab.  X.  1)  is  equal  to  d  divided  by  the  lengtu 


Ori 


CIRCULAR    CURVES. 


of  the  switched  rail.    If,  for  example,  the  rail  is  18  feet  in  lengih  and 
i  =  .42,  we  have  S  ==  P  20'. 


A.      Turnout  from  Straight  Lines. 

50.  ProfolCBll.     Given  the  radius  R  of  the  centre  line  of  a  tur*-oui 
(JiQ.  U),  to  fold  the  frog  angle  G  FM  =  F  and  the  chard  B  F. 


Solution.     Through  die  centre  E  draw  E  K  parallel  to  the  n     : 
track.      Draw   Ci/and   FK  perpendicular  to   E  K,  and  join  L  F. 
Then,  since  E  Fis  perpendicular  io  F?,J  and  F K  is  perpendicular  to 
FG,  the  angle  E  rK  =  G  FIvl  =  F;  and  since  E  B  and  B  H  are 
respectively  perpendicular  to  A  B  and  A  D,  the  angle  E  B  H  ^  DAB 

=  S.     Now  the  t'ianglc  E  F E gives  (Tab.  X.  2)  cos.  E  F  K  =   f-^ 

But  E  F,  the  radius  of  the  outer  rail,  is  equal  to  R  -\-  ^  g,  and 
FK=CH=Bn—  BC=B  E  cos.  E  B  H  —  B  C  =  ,R-\- ^  g) 
cos.  S  —  (g  —  d).  Substituting  these  values,  we  have  cos.  E  FK  ~ 
IK  +  iff)  COS.  5  -{g  —  d) 


B  +  is 


IS^ 


,or 

cos.  F  =  COS.  S  — 


9  —  ^ 

RTT9 


From  thin  formula  Fmay  be  found  by  the  table  of  natural  cosines 
To  adapt  it  to  calculation  by  logarithms,  we  may  consider^ —  d  to  be 
equal  to  (g — d)  cos.  S,  which  will  lead  to  no  material  error   since 


TURNOUT    FKOM    STRAIGHT    LINES.  33 

^  —  rf  is  very  small,  and  cos.  S  almost  equal  to  unity     The  value  of 
COS.  F  then  becomes 

1^  COS.  F  =  (^  —  ?  .9  +  c?)  COS.  S 

To  find  BF,  the  right  triangle  BCF  gives  (Tab.  X.  9)  BF  = 
BC 
Bin.  BFC-      ^"^  BC  =  y  —  d  and  the  angle  BF C  =  B FE 

CFE  -_  (900  _  LBEF)  -  (90°  -  F)  =  F  -  i  BEF     But 

BEF  -  Z?/.F  -  EBL  =  F -  S.      Therefore  BFC  =  F- 

■?  ^-^  ~  ^)  =  2  (^+  ^)-     Substituting  those  values  in  the  formula 
/or  B  F,  \vt  have 

sin.  '^{F+S)' 

By  the  above  formula;  tlie  columns  headed  /"and  i^i^in  Table  V 
are  calculated. 

Example.     Given  g  =  4.7,  d  =  .42,  S  =  1°  20',  and  R  =  .500,  to 

find  /"and  Z3  F.     Here  nat.  cos.  S  =  .999729,  g  —  d  =  4.28,  /2  +  ^^ 

=  .^Oo.SS,  and  4.28  -^  502  35  =  .008520.     Therefore  nat.  cos.  F  = 

999729  —  .008520  =  .991209,  which  gives  F=r  36'  10"      Next  to 

liud  OF, 

g  —  d  =  4.2S  0.631444 

H^+  S)  =4°28'5"       sin.  8.891555 

,  25  F=^  54.94  1.739889 

M     ProblCBta.      Given  the  frog  angle   GFM  =  F  {Jig.  14)^  to 
find^  the  radius  R  of  the  centre  line  of  a  turnout,  and  the  chord  B  F. 

Solution.      From    the    preceding    solution    Ave    have    cos.    F   = 
j-h2g)co3-  S—(g~d) 

K  +Ti  •      T^fjerefore  (R  +  ^  g)  cos.  F  =  {R  +  ^  g) 

COS.  ^  —  (g  —  d),  or 

^  R-^lg=  9-d 

cos.  aS'  —  COS.  F 

For  calculation  by  logarithms  this  becomes  (Tab.  X.  29) 

£^=  72  +  1^  = h  i9~d) 

sin.i(i^4-  ^')sin.  i(/^— ^y 

Having  thus  found  R  +  ^  g,  we  find  R  by  subtracting  ^  g.    B  F  u 
found,  as  in  the  preceding  problem,  by  the  formula 

7^  E  =  fJ  —  d 

3  sin.  !(/''+  6')  ■ 


S4 


CIRCULAR    CURVES. 


Example,     Given  g  =  4.7,  d  =  .42,  S  =  1'^  20',  and  F  =  7^  to  find 

«.     Here 

i  (^  —c/)  =  2.14  0.330414 

i  (F-f  5)  =  4'^  10' 

I  (Z'— 5)  =  2°  50' 


sin  8.861 283 
sin   8.G93998 


R  +  ^g  ^  595. 85 
.-.  R  =  593.5 


7.555281 
2.775133 


52.  Problem.  To  find  mechanically  the  proper  position  of  a  given 
frog. 

Solution.  Denote  the  length  of  the  switch  rail  by  /,  the  length  of  the 
frog  by/,  and  its  width  by  w.  From  B  as  a  centre  with  a  radius 
BH=  2/,  describe  on  the  ground  an  arc  G  H  K  {fig.  15),  and  from 
the  inside  of  the  rail  at  G  measure  G  H  =  2  d,  and  from  H  measure 
HK  such  that  HK :  B  H  =  ^  ic  :  f  ov  H K:  21  ^  ^  tv  :  f;  that  is, 

HK  =  y- .     Then  a  straight  line  through  B  and  the  point  K  will 
■-trike  the  inside  of  the  other  rail  at  F,  the  place  for  the  point  of  the 


tTOg.  For  the  angle  HB  K  has  been  made  equal  to  h  ^'  and  if  B  M 
be  drawn  parallel  to  the  main  track,  the  angle  MBH  is  seen  to  be 
equal  to  h  S.  Therefore,  MBK  =  BFC  =  ^  (F -[•  S),  and  this 
was  shown  (§  50)  to  be  the  true  value  of  B  F  C. 

53.  If  the  turnout  is  to  reverse,  and  become  parallel  to  the  main 
track,  the  problems  on  reversed  curves  already  given  will  in  general 
be  sufficient.  Thus,  if  the  tangent  points  of  the  required  curve  are 
fixed,  the  common  radius  may  be  found  by  §  40  If  the  tangent  point 
at  the  switch  is  fixed,  and  the  common  radius  given,  the  reversing 
oint  and  the  other  tangent  point  may  be  found  by  ^  37,  the  change 
)f  direction  of  the  two  tangents  being  here  equal  to  S.     Bur.  when  the 


TURNOUT    i^ROM    STRAIGHT    LINES. 


35 


frog  angle  is  given,  or  determined  from  a  given  first  radius,  and  the 
point  of  the  frog  is  taken  as  the  reversing  point,  the  radius  of  the  sec- 
ond portion  may  be  found  by  the  following  method. 

54.  Problem.  Given  the  frog  anjle  F and  the  distance  H  B  =  b 
(Jig.  16)  between  the  main  track  and  a  turnout,  tojind  the  radius  R'  of  the 
second  branch  of  the  turnout,  the  reversing  point  being  taken  opposite  F,  the 
fM}int  of  the  frog. 


Fig.  16 


Solution.  Let  the  arc  FB  be  the  inner  rail  of  the  second  branch, 
FG  =  R'  —  ^g  its  radius,  and  B  the  tangent  point  where  the  turnout 
becomes  parallel  to  the  main  track.  Now  since  the  tangent  FK  is  one 
side  of  the  frog  produced,  the  angle  HFK=  F,  and  since  the  angle 
of  intersection  at  iTis  also  equal  io  F,  BF K=  ^  F  {(^2,  II.) :  whence 

BFH=hF     Then  (^  es)  F  G  =  ^r^^^^  ,  or  R'  - 

:^-^^  (Tab.  X.  9),  or  i  Bi^=  ^-^ 


iBF 


But  BF 


sin.  iF-     sin 

6tituting  this  value  of  ^  B  F,  we  have 


^9  - 
Sub 


R' 


^  ^       sin.2 1  F 


In  measuring  the  distance  11 B  =  b,  it  is  to  be  observed,  that  tb« 
leidths  of  both  rails  must  be  included. 


36  CIRCULAR    CURVE'3. 

Example.     Given  6  =  6  2  and  i^  =  8^,  to  find  R'.     Here 

16  =  3.1  0.491362 

1  F  =  4^  sin.  8.84358.5 


1^^/.^=  44.44      1.647777 
i  F=  4'      sin.  8.843585 


/?'  — i^r  =  637.03      2.804192 
•.R'  =  639.43 


B.      Crossinrjs  on  Straight  Lines. 

55.  When  a  turnout  enters  a  parallel  main  track  by  a  second  switcn 
it  becomes  a  crossing.    As  the  switch  angle  is  the  same  on  both  tracks 
a  crossing  on  a  straight  line  is  a  reversed  curve  between  parallel  tar. 
gents.     Let  H D  and  iV/v  (fig.  17)  be  the  centre  lines  of  two  parallc 
tracks,  and  HA  and  B  /v  the  direction  of  the  switched  rails.     If  now 
the  tangent  points  A  and  D  are  fixed,  the  distance  A  B  ^  a  may  be 
measured,  and  also  the  perpendicular  distance  B  P  =  b  between  ?.;■? 
tangents  // P  and   B  K.     Tlicn  the  common  radius  of  the  crossing 
A  C  B  may  be  found  by  ij  33  ;  or  if  the  radius  of  one  part  of  the  cross- 
ing is  fixed,  the  second  radius  may  be  found  by  §  34.     But  if  both  frog 
nngles  are  given,  we  have  the  two  nidii  or  the  common  radius  of  a 
crossing  given,  and  it  will  then  be  necessary  to  determine  the  distance 
A  B  between  the  two  tangent  points. 

56.  Problem.  Given  the  perpendicular  distance  G  N=  b  (Jig.  17) 
between  the  centre  lines  of  two  parallel  tracks,  and  the  7Xidii  E  C  =-^  R  and 
CF  ^  R'  of  a  crossing,  to  find  the  chords  A  C  and  B  C 

Solution.  Draw  E  G  perpendicular  to  the  main  track,  and  A  L 
CM,  and  B  D  parallel  to  it.  Denote  the' angle  A  E  C  by  E.  Then, 
since  the  angle  A  E  L  =  AUG  =  S,  we  have  CE  L  =  E  -\-  S, 
and  in  the  right  triangle  C E  jV  (Tab.  X.  2),  CE  cos.  OEM  = 
R  cos.  {E  -]-  S)=^  EM=  EL  —  L  M.  But  EL  =  AE  cos.  A  EL 
=  R  cos.  S,  and  L  M  :  L'  M  =  A  C  :  B  C  Now  AC:  B  C  ^ 
E  C:  CF=  R:  R>.  Therefore,  L  M :  L'M  =  R:  R\  or  L  M :  LM 
■\-  L'M=  R:  R  +  R';  that  is,  L  M :  b  —  2d  =  R  :  R  -\- R',  whence 

L  M  =  "j^  ,    „,  - .     Substituting  these  values  of  E  L  and  L  Mm  the 

equation  for  R  cos.  {E  +  S),  we  have  R  cos.  {E  -\-  S)  =  R  cos.  S  — ■ 
R{b  —  2d) 
'  R4-  R'     ■> 


CROSSINGS    ON    STRAIGHT    LINES. 


S"? 


G^ 


/  n    1     e\  c       b  —  2d 

COS.  {L  +  S)  =  COS.  o  —  — 


A'  +  R' 

Having  thus  found  jE  +  S,  we  have  the  angle  E  and  also  its  equal 
VFB.     Then  (§  69) 

irr-        ^C=  2i2sin.  iJE;;  Z5  C  =  2  72' sin.  ^  ^. 

We  have  also  A  D  =  A  C -\-  B  C,  since  .4  C  and  Z?  C  are  in  the 
Ecme  straight  line  (§  32),  or  .d  C  =  2  (i?  +  72')  sin  ^  ^. 


Fig.  17. 


Whcu  the  two  radii  are  equal,  the  same  formulae  apply  by  making 
R'  =  R.     In  this  case,  we  have 

COS.  (E-\-S)  =  COS.  S  —    ~ '^    ; 

2  72 

AC=  BC=  2Rs\n.^E. 


Example.  Given  d  =  .42,  g  =  4.7,  5=1°  20',  6  =  11,  and  the  an- 
gles of  the  two  frogs  each  7°,  to  find  A  C  =  B  C  =^A  B.  The 
common  radius  72,  corresponding  to  F  =  7°,  is  found  (^  51)  to  be 
593..5.  Then  2  72=  1187,  6  —  2  (/ =  10.16,  and  10.16^-1187  = 
.008.56.  Therefore,  nat.  cos.  {E  -\- S)  =  .99973  —  .00856  =  .99117  ; 
whence  E -^  S  =  1°Z1'  15".  Subtracting  S,  we  have  E  =  6°  17'  15" 
Next 

2  72  =  1187  3.074451 

i  i?;  =  3°  8'  37^"     sin.  8.7.39106 


^  C=  65.1 


!  813557 


38 


CIRCULAR    CURVES. 


C.      Turnout  from  Curves. 

57.  Problem.      Given  the  radius  R  of  the  cadre  line  of  the  mair 
track  and  the  frog  angle  F,  to  determine  the  position  of  the  frog  by  means 
of  the  chord  B  F  {figs.  18  and  19),  and  to  find  the  radius  R'  of  the  cen 
tre  line  of  the  turnout. 


Fig.  18. 


Solution,  I.  When  the  turnout  is  from  the  inside  of  the  cunrt 
(fig.  18).  Let  A  G  and  CF  be  the  rails  of  the  main  track,  AB  the 
switch  rail,  and  the  arc  ^i^the  outer  rail  ot  the  turnout,  crossing  the 
inside  rail  of  the  main  track  sliF.  Then,  since  the  angle  E  FK  has  its 
sides  perpendicular  to  the  tangents  of  the  two  curves  at  F,  it  is  equal  to 
the  acute  angle  made  by  the  crossing  rails,  that  is,  E  F K  =  F.  Also 
E  B  L  ^  S.  The  first  step  is  to  find  the  angle  B  KF  denoted  by  K. 
To  find  this  angle,  we  have  in  the  triangle  B  FK{Tab.  X.  14),  BK-\- 
KF:BK—KF=  tan  ^  (B  FK  -{-  FB  K) :  tan.  ^  (B FK—  F  B  K). 
But  B  K  =  R  -\-  ^  g  —  d,  and  K  F  =^  R  —  ^  g.  Therefore,  B  K -^ 
KF  =  2R  —  d,  and  BK  —  KF  =■-  g  -  d.  Moreover,  B FK  = 
BFE  +  EFK=  BFE  +  F,  and  FBK=  EBF—EBK  = 
BFE  —  S.  Therefore,  BFK—FBK  =  F-^  S.  Lastly,  BFK 
-f-  FBK=  180°  —  K.     Substituting  these  values  in  the  preceding 


I  roportion.  we  have  2R  —  d:g  —d^  tan.  (90°  —  ^K):  tan.  |  (F- 


S), 


TURNOUT  FROM  CURVES.  39 

or  tan.  (90^  -  i  K)  =  il?.=3^^^ill±^ .    But  .an.  (90»  -  J  K) 

=  cot.  iii  =  i^rA'*' 

l^'  • .  tan.  h  K=  - — = .^  ~    ,  ,r,  ,    o>  • 

^  {2  11  — d)  tan.  J  (F+  /S) 

Next,   to  find   the  chord   B  F,  we  have,  in  the  triangle   B  F  C 
{T:ah.X.\2),BF=^/'^j.H^.     But  B  C  =  g  -  d,  and  BCF^ 

180°  —  FCK  =  180°  —  (90°  —  h  K)  =  90°  +  ^  A',  or  sin.  B  C  F 
=  COS.  I  K.  Moreover,  B  F  C  =^  hA^  -\-  S) ',  for  B F K  =  KFC 
-f  B  f''c,  and  F B  K  =  K  C F —BFC  =  KFC  —  BF C.  There- 
fore, B  F  K  -  FBK^2B  F  C.  But,  as  shown  above,  B  FK  — 
FB  K=  F+  S.  Therefore,  2  5 FC=  F+  5,  or  Z?FC=  ^  (F+  5). 
Substituting  these  values  in  the  expression  for  B  F,  we  have 

r^  ^  ,,  ^  jg  —  d)  COS.  |i^: 

•^  sin.H^+'5>')    ' 

Lastly,  to  find  R',  we  have  {k  ^^)  R'  -^  \g  =  E F  =  ^^J^  ^EF 
But  BE  F  =  BLF  —  EBL,  and  BLF  =  L  FK  +  L  ^F  = 
F  +  TT.     Therefore,  BEF  =F  -\-  K  —  S,  and 


sin.  ^(F+A^— 6^) 

II.  When  the  turnout  is  from  the  outside  of  the  curve,  the  preceding 
solution  requires  a  few  modifications.  In  the  present  case,  the  angle 
EFK'  =  F  (fig.  19)  and  EB  L  =  S.  To  find  K,  we  have  in  the 
triangle  B  F K,  K  F  -\-  B  K  :  KF  —  B  K  =  tan.  ^  (FB  K  + 
B F K)  :  tan.  i  (F C A'  —  i5 F A^.  But  KF=  R-{-lg,  and  B K 
=  R  —  i  g  +  d.  Therefore,  A"F  +  B  K  =--  2  R  +  d,  and  KF  — 
BK  =  g  —  d.  Moreover,  F B  A'  =  180°  —  F B  L  =  180°  — 
(EBF—EBL)  =  180°—  (E  BF  —  S),  and  BFK  =  180°  — 
BFK'  =  180^  —  (BFE  -\-  EFK')  =  180°  —  {E B F  +  F). 
Therefore,  FBK—B  FK  =  F  +  ^.  Lastly,  Fi3  K -\-  B  FK  = 
180  — K  Substituting  these  values  in  the  preceding  proportion,  we 
have  2R-\-d:  q  —  d=  tan.  (90°  —  ^K)  :  tan.  i  (F  +  S),  or 

,an.  (90°  -  i  A^)  =.  (2A±^^i^±^  .    But  tan.  ('90°  - 1  /T)  = 


tan.  h  K 


2 


g  —  d 


(2R  -\-d)  tan.  ^  (F+  .S) 


40 


CIRCULAR    CURVES. 


Next  to  find    B  F,  we   have,   in    the   triangle    B  T  -^     3  F  ^ 
B  C sin.  B  CF 


sin.  BFC 


But  BC  =  g  ~  d,  and  B  CF  =  90^ 

En 


AA,  or 


Fig.  19. 


sin.  BCF  =  COS.  ^K.  Moreover,  BFC=^(F+  S);  for  BFK 
=  KFC—BFC,and  FB  K=  KC  F-\- B  F  C  =  KF  C  +  BF  C. 
Therefore,  FBK— B  FK=2B  F  C.  But,  as  shown  above,  FBK— 
BFK=  F+  S.  Therefore,  2  BFC  =  F-}- S,ov B F C=^  {F-\- S). 
Substituting  these  values  in  the  expression  for  B  F,  we  have,  as  before. 

BF=  (ff  —  ^)  cos.  hK* 


},BF 


Sin.  1(^+5) 
Lastly,  to  find  R',  we  have  (§  GS)  R'  -\-  ^  fj  =r  E  F  = 


sin.  i  BEF 


Since  ^  Z  is  generally  very  small,  an  approximate  valu  iof  B  F  may  be  obtained 


By  making  cos.  ^  K  =  1.     Tliis  gives  B  F  =  — 


g-d 


- — ,  ;  T-,  r— c>  1  wbich  is  identical 
sm.  i  (F+  5)  ' 

with  the  formula  for  BF'm^  50.    Table  V.  will,  therefore,  give  a  close  approxima- 

4on  to  the  value  of  .B  F  on  curves  also,  for  any  value  of  F  contained  in  the  table 


TURNOUT  FROM  CURVES.  41 

Bvit  BEF  =  BLF  -  EBL,  and  BLF^LFK  —  LK  F  = 
p  _  A-.     Therefore,  DEF=F—K—  5,  and 


sin.^iF—  K—S) 


Example.  Given  g  =  4.7,  d  =  .42,  5  =  1°  20',  R  -  4583.75,  and 
F  =  7^,  to  Hnd  the  chord  B  Fund  the  radius  R'  of  a  turnout  from  the 
miside  of  the  curve.     Here 

q  —  cl  =  4.28  0.6.31444  0.631444 

2/2 +  (/=  9167.92  3.962271 

1  (/.^_}_  S)  =  4°  10'       tan.  8.862433  '         sin-  8.861283 

2.824704  1.770161 


1  7^-  ^  22'  1.8"  tan.  7.806740  cos.  9.999991 

GF=  58.905  1.770152 

2  0.301030 


|(/^  _  7v  —  ^')  =  2^  27'  58.2"  sin.  8.633766 


8.934796 


/i'  4- -I  ^  =  684.47  2.835356 

.-.R'  =  682.12 

58.  Problem.  To  Jind  mechanically  the  proper  position  of  a  given 
frog. 

■Solution.  Tlie  niotliod  here  is  similar  to  that  ah-eady  given,  when 
the  turnout  is  from  a  straight  line  (§  52).  Draw  B  .l/(figs.  18  and  19) 
parallel  to  /•'  C,  and  we  have  FBM  =  B  F  C  =  h  {F  +  S),  as  just 
shown  (§  57).  This  angle  is  to  be  laid  off  from  B  M ;  but  as  F  is  the 
point  to  be  found,  the  chord  F  C  can  be  only  estimated  at  first, .and 
B  M  taken  parallel  to  it,  from  which  the  angle  ^  (F -]-  S)  mi\y  be 
laid  off  by  the  method  of  §  52.  In  this  case,  however,  the  first  meas- 
ure on  the  arc  is  t/,  and  not  2  rf ,  since  we  have  here  to  start  from  B  i\f, 
and  not  from  the  rail.  Having  thus  determined  the  point  F  approxi- 
mately, B  M  may  be  laid  off  more  accurately,  and  F  found  anew. 

59.  When  frogs  are  cast  to  be  kept  on  hand,  it  is  desirable  to  have 
them  of  such  a  pattern  that  they  will  fall  at  the  beginning  or  end  of  a 
certain  rail;  that  is,  the  chord  B  F  is  known,  and  the  angle  F  is  re- 
quired. 


l2  CIRCULAR    CURVES. 

Problem*  Given  the  position  of  a  frog  by  means  of  the  chord  B  F 
[figs.  14,  18,  and  19),  to  determine  the  frog  angle  F. 

g  —  d 
Solution.     The  formula   B  F  =  gin  ^^(F  -\-  S) '  ^^^^^  ^^  exact  on 

straight  lines  (§  50),  and  near  en'jugh  on  ordinary  curves  (§  57,  note), 
gives 

1^  sin.^(F+^)=5:^. 

By  this  formuUi  ^  {F  -\-  S)  may  be  found,  and  consequently  F. 

60.  Problem.  Gii^en  the  radius  R  of  the  centre  line  of  the  main 
tracks  and  the  radius  R'  of  the  centre  line  of  a  turnout,  to  find  the  frog 
angle  F,  and  the  chord  B  F  {figs.  18  and  19). 

Solution.  I.  When  the  turnout  is  from  the  inside  of  the  curve 
(fig.  18).  In  the  triangle  BE  Kfind  the  angle  B  E  K  and  the  side  E  K. 
For  this  purpose  we  have  B E  =  R'  +  h g,  B K  =  R -\-  ^ g—  d,  &nd 
the  included  angle  E  BK  =  S.  Then  in  the  triangle  E FK  we  have 
E  K,  as  just  found,  E  F  =  R'  -{-  ^ g,  and  F K  =  R—  ^ g  The  frog 
angle  EFK  =  F  .nay.  therefore,  be  found  by  formula  15,  Tab.  X., 
which  gives 


tan.  A  F  = 


_      l(s-6)(5-c) 


V 


s  {s  —  a) 

where  s  is  tiie  half  sum  of  the  three  sides,  a  the  side  E  K,  and  b  and  c 
the  remaining  sides. 

Find  also  in  the  triangle  EFK  the  angle  F E  K,  and  we  have  the 
angle  BE  F  =  BEK  -  FEK.  Then  in  the  triangle  B E F  we 
have  (§69) 

1^^  BF=2{R' +  ^g)  s'm.^ BE F* 

II.  AVhen  the  turnout  is  from  the  outside  of  the  curve  (fig.  19).  In 
the  triangle  B  E  K  find  the  angle  BEK  and  the  side  EK  For  this 
purpose  we  have  B  E  =  R'  -\-  ^  g,  B  K  =  R  —  ^  g  -\- d,  and  the  in- 
cluded angle  E  BK=  180=  —  aS.  Then  in  the  triangle  E  FK  vff 
Iiave  E K,  as  just  found,  E  F  =  R'  -\-  ^  g,  and  F K  =  R+  ^  g.  The 
angle  EFK  may,  therefore,  be  found  by  formula  15,  Tab.  X.,  which 

gives  tan.  ^EFK  =  V^' 7(5-0]^^  •     ^"^  ^^'°  '^"^'^^  ^  ^^'  =  ^ 


*  The  value  of  B  F  maj'  be  more  easily  found  by  the  approximate  formula  B  F  = 
,  and  generally  with  sufficient  accuracy.     See  note  to  §  57.     This  re- 


nin. i{F+  S) 

mark  applies  also  to  B  F  in  the  second  part  of  this  solution. 


TURNOUT  FROM  CURVES.  43 

^  ISO''  —  EFK.    Therefore  ^F  =  90°  — ^EFK,  and  cot  ^  F  =• 
tm.  ^EFK',  . 

t^  . • .  cot.  ^F=  \  5^ — ^— — T — '  » 

•^  ^  ^       s  (s  —  a) 

where  s  is  tlie  half  sum  of  the  three  sides,  a  the  side  ^  K,  and  6  and  c 
the  remaining  sides. 

Find  also  in  the  triangle  EFK  the  angle  FE  K,  and  ive  have  the  angle 
BE  F=  FE  K  —  BE  K.    Then  in  the  triangle  BE  F  we  have  (§  69) 

13^  BF^2{R'  +  ^g)sm.^BEF 

Example.  Given  g  =  4.7,  d  =  .42,  5=1°  20>,  R  =  4583.75,  and 
/{>  r=  682.12,  to  find  F  and  the  chord  Z?  Fof  a  turnout  from  the  outsida 
of  the  curve.  Here  in  the  triangle  i3 £ /v  (fig.  19)  we  have  BE  = 
^,  ^  i  ^  ^  684.47,  BK=R  —  kf}  +  d  =  4581  82,  and  the  angles 

BEK+  BKE  =  S=l°  20'.     Then 

BK— BE  =  3807.35  3.590769 

^{BEK+BKE)=  40'  tan.  8.065806 

1.656575 

BK-\-  BE  =  5266.29  3.721505 

^  [BEK—  BKE)*  =  29.6029'      tan.  7.935070 

.'.  BEK=  l""  9.6029' 

„  ^        BK  sia  EBK  .         „  „ 

EK\s  now  found  by  the  formula  EK=      sin  BEK^  '  ^^'  '^S-  ^  ^ 

=  log.  4581.82  +  log.  sin.  178°  40'  —  log.  sin.  1°  96029'  =  3.721491, 
whence  £ir=  5266.12. 

Then  to  find  F,  we  have,  in  the  triangle  EFK,  s  =  ^  (5266.12  -f- 
684.47  +  4586.10)  =  5268.34,  s  —  a  =^  2.22,  s  —  6  =  4583.87,  and 

s-   c  =  682.24. 

s_6  =  4583.87  3.661233 

s  —  c  =  682.24  2.833937 


s  =  5268.34   3.721674 
—  a  =  2.22     0.346353 


6.495170 

4.068027 

2)^27T43 

^F=3°  30'  cot. 72135 71 

.•.F=  7° 


•  This  angle  and  the  sine  of  1°  9  6029'  below,  are  found  by  the  method  given  in 
•onnection  with  Table  XIII.  If  the  ordinary  interpolations  had  been  used,  wa 
should  have  found  F  =  7'^  7',  whereas  it  should  be  7^,  since  this  example  is  tha 
•inverse  of  that  in  §  57. 


14 


CIRCULAR    CURVES. 


To  find  FEK,  we  have  s  as  before,  but  as  a  is  here  the  side  FR 
opposite  the  angle  sought,  we  have  s  —  a  =  682.24,  s  —  h  =  458.'?  87, 
and  s  —  c  =  2. 22.  Then  bv  means  of  the  logarithms  just  used,  we 
find  ^FEK=  3^  2'  45".  Sul)tnicting  ^  B  E  K  =  W  48",  we  have 
^BEF  ^  2°  27'  57".  Lastly.  BF  =  1368  94  sin.  2^  27'  57"  = 
58.897. 

The  formula  ^ J^  =  sm.t{F+  S)  (§  5"'  "ote)  would  give  BF  = 
58  906,  and  this  value  is  even  nearer  the  truth  than  that  just  found, 
owing,  however,  to  no  eiTor  in  the  formulfe,  but  to  inaccuracifs  inci- 
dent to  the  calculation. 

61.  If  the  turnout  is  to  reverse,  in  order  to  join  a  track  parallel  to 
the  main  track,  as  A  CB  (fig.  20),  it  will  be  necessary  to  determine 
the  reversing  points  C  and  B.  These  points  will  be  detennined,  if  we 
find  the  angles  A  E  C  and  B  F  C,  and  the  chords  A  C  and  CB. 


62  Problem.  Given  the  radius  D  K  =  R  {Jig  20)  of  the  centrt 
line  of  the  main  truck  the  common  radius  E  C  =  CF  =  R'  of  the  centre 
line  of  a  turnout,  and  the  distance  B  G  =  b  between  the  centre  lines  of  the 
^parallel  tracks,  to  find  the  central  angles  A  E  C  and  B  F  C  and  the  chorda 
A  C  and  EC. 


Solution.     In  the  triangle  A  E  K  fitrd  the  angle  AEK  and  the  side 


CROSSINGS    ON    CURVES.  i5 

e  K  For  tliis  purpose  we  have  AE  =  R',  A  K  =  R  —  d,  and  tlic 
included  angle  E  A  K  =-  S.  Or,  if  the  frog  angle  has  been  previously 
calculated  by  §  GO,  the  values  of  A  E  K  and  E  K  are  already  known.* 
Find  in  the  triangle  EFK  the  amjles  E  FKand  F E  K  For  this 
purpose  we  have  E  K^  as  just  found,  E  F  ^  2  A",  and  FK  =  A  -^- 
R'  —  h.  Then  AE  C  =  AEK  —  FEK,  and  BFC  ^  E FK. 
Lastly,  (§69) 

^^    AC^2Rs\n^AEC;     C  B  =  2  R' sin.  ^  B  F  C. 

This  solution,  with  a  few  obvious  modifications,  will  apply,  when 
the  turnout  is  from  the  outside  of  a  curve. 

D.     Crossings  on  Curves. 

63.  When  a  turnout  enters  a  parallel  main  track  by  a  second  switch, 
■  t  becomes  a  crossing.  Then  if  the  tangent  points  A  and  B  (fig.  21) 
are  fixed,  the  distance  A  B  must  be  measured,  and  also  the  angles 
which  A  B  makes  with  the  tangents  at  A  and  B.  The  common  ra- 
dius of  the  crossing  may  then  be  found  by  §  40 ;  or  if  one  radius  of  the 
crossing  is  given,  the  other  may  be  found  by  \  38.  But  if  one  tangent 
point  A  is  fixed,  and  the  common  radius  of  the  crossing  is  given,  it 
will  be  necessary  to  determine  the  reversing  point  C  and  the  tangent 
point  B.  These  points  will  be  determined,  if  we  find  the  angles  AEC 
vind  B  F  C,  and  the  chords  A  Cand  C  B. 

64.  Problem.  Given  the  radius  DK=  R  {Jig-  21)  of  the  cetitte 
line  of  the  main  track,  the  common  radius  E  C  =  C  F  =  R'  of  the  centre 
line  of  a  crossing,  and  the  distance  D  G  =  b  between  the  centre  lines  of  the 
parallel  tracks,  to  find  the  central  angles  AE  C and  B  F  C and  the  chords 
A  Cand  CB. 

Solution.  In  the  triangle  AEK  find  the  angle  AE  K  and  the  side 
E  K.  For  this  purpose  we  have  A  E  =  R',  A  K  =  R  —  d,  and  the 
included  angle  E  Ax  K  =  S. 

Find  in  the  triangle  B  FK  the  angle  B  F K  and  the  side  F K.  For 
this  purpose  we  have  B  F  ^  R',  B  K=  R  —  h  +  d,  and  the  included 
&ng\QFBK=  180=^  —  6'. 

Find  in  the  trianale  EFK  the  angles  F E  K and  EFK.     For  this 


*  The  triangle  AEK  does  not  correspond  precisely  with  BEKm^  ^,  A  being 
on  the  centre  line  and  B  on  the  outer  rail  ;  but  the  difference  is  too  slight  to  affect 
the  calculations. 


16 


CIRCULAR    CURVES. 


purpose  we  have  E  K  and  FK  a.s  just  found,  and  E  F  —-  2  W.  rhet> 
AEC  =^  AEK—  FEK,  and  BFC^EFK—B  FK.  Lastlv 
(§  69,) 

AC=^2R<  sm.hAE  C',     CB  ==  2  R' sin.  ^  BF  C. 
D 


Fig.  21. 


Article  IV,  —  Miscellaneous  Problems. 

65,  Problem.  Given  A  B  =  a  [Jig.  22)  and  the  perpendicular 
B  C  =  b,  to  Jind  the  radius  of  a  curve  that  shall  pass  through  C  and  the 
tangent  point  A. 

Solution.  Let  0  be  the  centre  of  the  curve,  and  draw  the  radii  A  0 
and  C  0  and  the  line  CD  parallel  to  A  B.  Then  in  the  right  triangle 
COD  we  have  0  C^  =  CD""  +  OD^  But  0  C  =  R,  CD  =  a,  and 
OD  =  AO  —  AD  =  R  —  b.  Therefore,  R""  =  a"" -{-  {R  —  6)»  = 
a^  +  R^  —  2  Rb  -\-  b\  or  2  Rb  =  a^  -{-  b^ ; 

2  b 

Example.     Given  a  =  204  and  b  =  24,  to  find  R.     Here  R  »- 

204-2         24 
2X-24  +  2   =  «67  +  12  =  879. 


iillSCELLANEOUS    PROBLEMS. 


47 


C6.  Corollary  1.     If  R  and  b  are  given  to  find  A  B  =  a,  that 
vs,  to  determine  the  tangent  point  from  which  a  curve  of  given  radius 


most  start  to  pass  through  a  given  point,  we  have  (§65)  2Rb  = 
fl«-f  i^ora'  =  2Rb  —  b^; 


.'.a  =  ^b  {2R  —  b). 

Example.  Given  6  =  24  and  72  =  879,  to  find  a.  Here  o  =- 
/94  (1758  —  24)  =  ^  41616  =  204. 

67.  Corollary  2.  If  R  and  a  are  given,  and  b  is  required,  we 
have  (§65)  2  Rb  =  a^  +  6^  or  6«  —  2Rb  =  —a}.  Solving  this 
equation,  we  find  for  the  value  of  b  here  required, 


b  =  R  —  ^R-  —  a\ 

68.  Problem.  Given  the  distance  AC  =  c  [Jig.  22)  and  the  an- 
gle B  A  C  ^  A,  to  find  the  radius  R  or  deflection  angle  D  of  a  curve,  that 
fhall  pass  through  C  and  the  tangent  point  A. 

Solution.  Draw  0  E  perpendicular  to  A  C  Then  the  angle  AOE 
^^A0C  =  BAC=A{(j2,  III.),  and  the  right  triangle  A  OEgWos 

(Tab.X.9)^0  =  3j^^i^; 

•   R-     ^^ 
Sin.  A 


To  find  Z),  we  have  (§  9)  sin.  D  = 
•nst  found,  we  have  sin.  Z)  =  50  -^• 


^ .     Substituting  for  R  its  value 

he 
sin.  A  ' 


48  CIRCULAR    CURVES. 

c 

Example.     Given  c  =  2S5.t  and  ^l  =  5°,  to  find  R  and  D.     Heix. 

^,  142.7  ,^„,„  ,     .       ^         iOOsin.  50        sin.  5-^ 

^'  =  ^75"^  =  163/. 3  ;  and  sin.  D  =  -^g^-  =  2So4   =  s'"-  ^    "^^ 

or  D  =  1  o  45'. 

69.  Problem.  Given  the  radius  R  or  the  deJiecUon  amjle  D  of  a 
curve,  and  the  angle  B  A  C  =  A  {Jig.  22),  made  by  any  chuid  with  the 
tangent  at  A^  to  find  the  length  of  the  chord  A  C  ■=  c. 

he 
Solution.    If  R  is  given,  we  have  (§  68)  R  =  ^^— j  ; 

.-  .c  =  2  R  sin.  ,1. 


Ti»  -n,  •       •                  1           ,,   ^^v     •       r>         100  sin.  A 
n  D  IS  given,  we  have  (§  68)  sin.  D  ^ — 


100  sin.  A 
c  == 


sin.  D 

This  formula  is  useful  for  finding  tlic  length  of  chords,  when  a  curve 
is  laid  out  by  points  two,  three,  or  more  stations  apart.  Thus,  suppose 
that  the  curve  ^  Cis  four  stations  long,  and  that  we  wish  to  find  the 
length  of  the  chord  A  C.  In  this  case  the  angle  A  =  A  D  and  c  = 
100  sin.  4  D 


sin.  D 


Bv  this  method  Table  II.  is  calculated. 


Example.  Given  R  =  2455.7  or  Z>  =  1°  10',  and  .1  =  4°  40',  to 
find  c.     Here,  by  the  first  formula,  c  =^  4911.4  sin.  4°  40'  =  399.59. 

,^      ,  ,  ^  ,  100  sin.  43  40' 

Isy  the  second  formula,  c  —     gin  \o  iq'     =  399.59, 

70.  ProblCDll.  Given  the  angle  of  intersection  K  C  B  =  1  [fig.  23), 
and  the  distance  CD  =  h  from  the  intersection  point  to  the  curve  in  the 
direction  of  the  centre.,  to  find  the  tangent  A  C  =  T,  and  the  radius  A  G 
=  R. 

Solution.     In  the  triangle  ^  D  C  we  have  sin.  CA  D  :  sin.  A  D  C  =^ 

CD:  AC.     Bnt  CAD  =  ^AOD  =  ili^  2,  III.  and  VI.),  and  as 

the    sine   of  an   angle   is  the   same  as  the   sine   of  its   supplement, 

sin.  A  D  C  ==  sin    A  D  E  =  cos.  DA  E  =  cos    4  /.     Moreover,  CD 

=  b  and  A  C  =  T.     Substituting  these  values  in  the  prectrding  pro- 

b  cos,  -^  ^ 
portion,  we  have  sin.  ^  I :  cos.  ^  I  =  b  :  T,  or  T  =  ^.^  \*j^  ;  whence 

(Tab.  X.  33) 


MISCELLANEOUS    PROBLEMS. 


19 


^-  T  =h  cot.  \  I. 

To  find  R,  we  have  (§  5)  R  =  T  cot.  ^  I.     Substit  iting  for  T  ifc 
falue  just  found,  wc  have 

^"  R  =  b  cot.  ^  7  cot.  ^  2 


Fig.  23. 


hxample.     Given  7  =  30°,  6  =  130,  to  find  Tan!  R.     Here 


h  =  130 
^7=7°  30' 

7'  =  987.45 
17=  15° 

72  =  368.5.21 


2.113943 
cot.  0.880571 


2.994514 
col.  0.571948 


3.566462 


7 1 .  Problem.     Given  the  angle  of  intersection  KC  B  =  1  [Jig.  23 ). 

%nd  the  tangent  A  C  =  T,  or  the  radius  A  0  =  R,  to  find  C  D  -^  b. 

Solution.    If  T  is  given,  we  have  (§  70)  T  =  h  cot.  ^  7,  or  6  = 
T 


lot  i/' 


.•.h=  r  tan.  17. 

If  R  is  given,  we  have  (§  70)  R  =  b  cot.  ^7  cot.  |^7,  or  6 
R 


eot  ^  Jcot.  i  /  ' 


.'.b  =  R  tan.  ;J  7  tan.  ^  7. 


50 


CIRCULAR    CURVES. 


Example.  Given  /=  27°,  T=  600  or  7^  =  2499  lb,  to  fin.l  I 
Here  b  =  600  tan.  6°  45'  =  71  01,  or  i  =  2499.18  tan.  6°  45 
tan.  13°  30'  =  71.01. 

1 

72.  Problem.  Given  the  angle  of  intersection  I  of  two  tangent 
A  C  and  D  C  (fg.  24)  to  find  the  tangent  point  A  of  a  curve,  that  shed 
pass  through  a  point  E,  given  by  C D  ==  a,  D  E  =^  b,  and  the  angle  CD  E 


Eig.  24 


Solution.  Produce  DE  to  the  curve  at  G,  and  dra^7  C  0  to  the  cen- 
tre 0.  Denote  DFbyc.  Then  in  the  right  triangle  CDF  we  have 
(Tab.  X.  U)  DF=  CD  cos.  CDF,  or 


c  =  a  cos. 


Denote  the  distance  A  D  from  D  to  the  tangent  point  by  x.  Then,  by 
Geometry,  x^  =  D  E  X  D  G.  But  D  G  =  D  F -\-  FG  =  DF  + 
EF=2DF— DE  =  2c  —  b.     Therefore,  x^  =  b{2c  —  b),  and 


5^"  x  =  ^b{2c  —  b). 

Having  thus  found  A  Z),  we  have  the  tangent  AC  =  AD  -{•  DC 
=  X  -\-  a.     Hence,  R  ox  D  may  be  found  (^  5  or  §  11). 

If  the  point  E  is  given  by  £^^and  Ci/ perpendicular  to  each  other, 
a  and  b  may  be  found  from  these  lines.    For  a  =  C  H  -\-  DH  ^ 

(75"+ JE;77cot.  iZ(Tab.  X.  9).  and6  =^DE  =  ^^i- 


MISCELLANEOUS    PROBLEMS. 


5i 


Example.  Given  I  =  20°  16',  a  =  600,  and  6  =  80,  to  find  x  and 
H.  Here  c  =  600  cos.  10°  8'  =  59064,  2  c  -  6  =  1101. 28,  and  x  = 
ySO  X  110^28  =  296.82.  Then  T  =  600  +  296.82  =  896.82,  and 
R  =  896.82  cot.  10°  8'  =  5017.82. 

73.  Problem.  Given  the  tangent  A  C  {Jig.  25),  and  the  chora 
A  By  uniting  the  tangent  points  A  and  B,  to  Jind  the  radius  A  0  --  R. 


Fig.  25 


Solution.  Measure  or  calculate  the  perpendicular  CD.  Then  if  CZ) 
be  produced  to  the  centre  0,  the  right  triangles  AD  C  and  CA  0, 
having  th3  jungle  at  G  common,  are  similar,  and  give  CD  :  A  D  = 
AC:  A  0,  or 

^^A^XAC 
CD 

If  it  is  inconvenient  to  measure  the  chord  A  B,  a  line  E  F,  parallel 
to  it,  may  be  obtained  by  laying  off  from  C  equal  distances  CE  and 
CF.     Then  measuring  E  G  and  G  C,  we  have,  from  the  similar  tri- 

GEXAC 
%ng\esE  GCand  CAO,  CG:GE  =AC:AO,orR= — ^G — * 

Example.     Given  ^  C  =  246  and  AD  =  240,  to  find  R.     Here 

240  X  246 
VD  =  54,  and  R  =  -    ^'^=  1093.33. 


52  CIRCULAR    CURVES. 

74.  Problem.  Given  the  radius  AO  =  R  [foj  25),  to  find  :ht 
tangent  A  C  =  J-  of  a  curve  to  unite  two  straight  lines  given  on  the  ground 

Solution.  Lay  off  from  the  intersection  C  of  the  given  straight  lines  any 
equal  distances  CL  and  CF.  Draw  the  pe7-pendictdar  C  G  to  the  mid- 
dle of  E  F,  and  measure  G  E  and  C  G.  Then  the  right  triangles 
E  G  Cand  C  A  0,  having  the  angle  at  C  common,  are  similar,  and 
give  GE:  CG  =  AO:  AC,  or 

EF-  r^__CGx  AO 

GE 

By  this  problem  and  the  preceding  one,  the  radius  or  tangent  points 
of  a  curve  mav  be  found  without  an  instrument  for  measuring  angles. 

Example.     Given  R  =  1093|,  G  E  =  80,  and  C  G  =  18,  to  find  '/'. 

18X1093^ 
Here  F  = gQ =  246. 

75.  Problem*  To  find  the  angle  of  intersection  I  of  two  straight 
lines,  when  the  point  of  intersection  is  inaccessible,  and  to  determine  the  tan- 
gent points,  when  the  length  of  the  tangents  is  given. 

Solution.  I.  To  find  the  angle  of  intersection  i  L.ct  A  C  and  C  I' 
(fig.  26)  be  the  given  lines  Sight  from  some  point  A  on  one  line  lo  a 
point  B  on  the  other,  and  measure  the  angles  CAB  and  T B  V.  These 
angles  make  up  the  change  of  direction  in  passing  from  one  tangent  to 
the  other.  But  the  angle  of  intersection  (§  2)  shows  the  change  of  di- 
rection between  two  tangents,  and  it  must,  therefore,  be  equal  to  the 
sum  of  C  A  B  and  T  B  V,  that  is, 

t^  1=  CAB-^  TBV 

But  if  obstacles  of  any  kind  render  it  necessary  to  pass  from  A  C  to 
B  Fby  a  broken  line,  as  A  D  E  F  B,  measure  the  angles  C  A  D,  N D  E, 
P  E  F,  RFB,  and  S  B  V,  observing  to  note  those  angles  as  mimts  which 
are  laid  off  contrary  to  the  general  direction  of  these  angles.  Thus  the 
general  direction  of  the  angles  in  this  case  is  to  the  right;  but  the 
angle  P EF  lies  to  the  left  oi  D E  produced,  and  is  therefore  to  be 
marked  minus.  The  angles  to  be  measured  show  the  successive  changes 
of  direction  in  passing  from  one  tangent  to  the  other.  Thus  C  A  D 
6hov/s  the  change  of  direction  between  the  first  tangent  and  A  D, 
ND  E  shows  the  change  between  A  D  produced  and  D  E,  P  E  F  the 
change  between  DE  produced  and  E  F,  R  F B  the  change  between 
£'F produced  and  FB,  and,  lastly,  SB  Fthe  change  between  B  F  ])ro- 


MISCELLANEOUS    PROBLEMS. 


53 


duccd  and  the  second  tangent.  But  the  iing^lc  of  intersection  (§  2) 
shows  the  change  of  direction  in  passing  from  one  tangent  to  another, 
and  it  must,  therefore,  be  equal  to  the  sum  of  the  partial  changes 
naeasuved,  that  is, 


13^ 


/  =  CA  D  -\-  y  DE  -  PEF-^  II FB  +  SB  V. 


Fig.  26 


II.  To  determine  the  tangent  points.  This  will  be  done  if  we  find 
the  distances  .1  Cand  B  C;  for  then  any  other  distances  from  Cmay 
be  found.  It  is  supposed  that  the  distance  A  B,  or  the  distances  A  Z), 
DE,  E  F,  and  FB  have  been  measured. 

Tf  one  line  A  B  connects  A  and  B.  Jind  A  C  and  B  C  in  the  triangle 
ABC.     For  this  purpose  we  have  one  side  A  B  and  all  the  angles. 

Jf  a  broken  line  A  D  E  F  B  connects  A  and  B,  let  fall  a  perpendicular 
B  G  from  B  upon  A  C,  produced  if  necessary,  and  find  A  G  and  B  Q 
hy  the  usual  method  of  working  a  traverse.  Thus,  if  A  C  is  taken  as  a 
meridian  line,  and  D  /v,  E  L,  and  FM  are  drawn  parallel  to  A  C,  and 
D  H,  E  K,  and  FL  are  drawn  parallel  to  B  G,  the  difference  of  lati- 
tude A  G  is  equal  to  the  sum  of  the  partial  differences  of  latitude  A  H. 
D  K,  EL,  and  FM,  and  the  departure  B  G  h  equal  to  the  sum  of  the 
partial  departures  D  II,  E  K,  F  L,  and  B  HI.  To  find  these  partial 
differences  of  latitude  and  departures,  we  have  the  distances  A  I),  DE, 
E  F,  and  F  B,  and  tiie  bearings  may  be  obtained  from  the  angles 
already  measured.  Thus  the  bearing  of  yl  Z)  is  C  A  D,  the  bearing  of 
DE  is  KDE  =  KDN+  NDE  =^  C  A  D  -\-  NDE,  the  bearing 
of  jB F  is  LEF  =  LEP—  PEF^  KDE—  PEF,  &nA  the 


54 


CIRCULAR    CURVES. 


bearing  oi F B  is  MFB  =  MFR  -{-  RFB=^  LEF  +  RFB;  that 
is,  the  bearing  of  each  line  is  equal  to  the  algebraic  sum  of  the  preced 
ing  bearing  and  its  own  change  of  direction.     The  differences  of  lati- 
tude and  the  departures  may  now  be  obtained  from  a  traverse  table, 
or  more  correctly  by  the  formulis : 

DiiF.  of  lat.  =  dist.  X  cos.  of  bearing  ;   dep.  =  dist.  X  sin.  of  bearing 

Thus,  AH=  AD  cos.  CAD,  and  DU=AD  sin.  CA  D. 

Having  found  A  G  and  B  G,  we  have,  in  the  right  ti'iangle  B  G  C, 

(Tab.  X.  9)  GC  =  B  G  cot.  B  C  G,  and  BC  =  ^^^-q  ■  But 
BCG=180°  —  I.  Therefore,  cot.  BCG  =  —  cot.  /,  and  sin.BCG 
=  sin.  /.  Hence  G  C  =-  —  B  G  cot.  7,  and  BC  =  ^^^77 .  Then, 
since  A  C  =  A  G  -\-  G  C,  we  have 


AC=AG  —  BG  cot.  /; 


BC 


BG 


sm. 


When  /is  between  90°  and  180°,  as  in  the  figure,  cot.  /is  negative, 
and  — B  G  cot.  I  is,  therefoi-e,  positive.  When  /  is  less  than  90°,  G 
will  fall  on  the  other  side  of  / ;  but  the  same  formula  for  A  C  wil  still 
apply  ;  for  cot.  /  is  now  positive,  and  consequently,  — B  G  cot.  /  is 
negative,  as  it  should  be,  since,  in  this  case,  A  C  would  equal  A  G  mi 
mis  G  C. 

Example.  Given  A  D  =  1200,  DE  =  350,  E F  ==^  300,  F B  =^ 
310,  CAD==  20°,  NDE  =  44°,  PE  F  =.  —  25°,  R  FB  =  31°. 
and  SB  V  ^  30°,  to  find  the  angle  of  intersection  /,  and  the  distance? 
A  C  and  B  C. 

Here  7  =  20°  +  44°  —  25°  +  31°  +  30°  =  100°.  To  find  A  G 
and  B  G,  the  work  may  be  arranged  as  in  the  following  table :  — 


Angles  to 
the  Right. 

Bearings. 

Distances. 

N. 

£. 

0 
20 

44 

—25 

31 

N.  20  E. 
64 
39 
70 

1200 
3.50 
300 
310 

1127.63 
153.43 
233.14 
106.03 

410.42 
314.58 

188.80 
291.30 

1620.23 

1205.10 

The  first  column  contains  the  observed  angles.    The  second  contains 
the  bearings,  which  are  found  from  tne  angles  of  the  first  column,  iv 


MISCELLANEOUS    PROBLEMS. 


55 


the  manner  already  explained.  A  Cis  considered  as  running  north 
from  A,  and  the  bearings  are,  therefore,  marked  N.  E.  The  other  col- 
umns require  no  explanation.  "We  find  A  G  =  1620.23,  and  B  G  = 
1205.10.  Then  GC  =  —  BG  cot.  I  =  —  1205.1  X  cot.  100°  =- 
212.49.  This  value  is  positive,  because  it  is  the  product  of  two  nega- 
tive factors,  cot.  100°  being  the  same  as  —cot.  80°,  a  negative  quanti- 
ty.     Then  AC=  AG  +  GC=  1620.23  +  212.49  =  1832.72,  and 

BC  =  -. — ^bn  =  1223  69.      Having  thus  found  the  distances  of  A 
sin.  1UU-'  ° 

and  B  from  the  point  of  intersection,  we  can  easily  fix  the  tangent 
points  for  tangents  of  any  given  length. 

76.  Problem.      To  Uuj  out  a  curve,  when  an  obstruction  of  any  kind 
prevents  the  use  of  the  ordinarij  methods. 


^ig.  27 


Solution.  First  Method.  Suppose  the  instrument  to  be  placed  at 
A  (fig.  27),  and  that  a  house,  for  instance,  covers  the  station  at  B,  and 
also  obstructs  the  view  from  A  to  the  stations  at  D  and  E.  Lay  off 
from  A  C,  the  tangent  at  yl,  such  a  multiple  of  the  deflection  angle  Z), 
iis  will  be  sufficient  to  make  the  sight  clear  the  obstruction.  In  the 
figure  it.  is  supposed  that  4  Z)  is  the  proper  angle.  The  sight  will  then 
pass  through  F,  the  fourth  station  from  A,  and  this  station  will  be  de- 
termined by  measuring  from  A  the  length  of  the  chord  A  F,  found  by 


56 


CIRCULAR    CURVES. 


§  69  or  by  Table  II.  From  the  station  at  i^  the  stations  at  D  and  E 
may  afterwards  be  fixed,  by  laying  off  the  proper  deflections  from  the 
tangent  at  F. 

Second  Method.  This  consists  in  running  an  auxiliary  curve  paral 
lei  to  the  true  curve,  either  inside  or  outside  of  it.  For  this  purpose 
lay  off  perpendicular  to  A  C,  the  tangent  at  A,  a  line  A  A'  of  any  con 
venient  length,  and  from  A'  a  line  A'  C  parallel  to  A  C.  Then  A'  C' 
is  the  tangent  from  which  the  auxiliary  curve  A<  E'  is  to  be  laid  off. 
The  stations  on  this  curve  are  made  to  correspond  to  stations  of  100 
feet  on  the  true  curve,  that  is,  a  radius  through  B'  passes  through  Zj,  a 
radius  through  D'  passes  through  D,  &c.  The  chord  .4'  B'  is,  tlicre- 
fore,  parallel  to  A  B,  and  the  angle  C  A'  B'  =  CAB;  tliat  is,  the  de- 
flection angle  of  the  auxiliary  curve  is  equal  to  that  of  the  true  curve 
It  remains  to  find  the  length  of  the  auxiliary  chords  A' B',  B'  D',  &c 
Call  the  distance  A  A'  =  h.  Then  the  similar  triangles  ABO  and 
A'  B>  0  give  A  0  :  A'  O  =  A  B  :  A'  B',  or  R  :  R  —  b  =  100  :  A'  B>. 

Therefore  A<  B<  -  ^^^<^~'^  _  i  no       ^^^  *       tp  .i  -r 

j-ueicrore,  ^  ij    —  ^  =  100 —  — ^  .     If  the  auxihary  curve 

were  on  the  outside  of  the  true  curve,  we  should  find  in  the  same  way 

.-l'  B'  ^  100  4-  -^  .  It  is  well  to  make  h  an  aliquot  part  of  R  ;  foi 
the  auxiliary  chord  is  then  more  easily  found.     Thus,  if  n  is    anv 

whole  number,  and  we  make  6  =  -  ,  we  have  A'  B'  =  100  ±  ^%^ 

=  100  ±  —  .     If,  for  example,  ^  =  Jq^  ,  we  have  ??  =  100,  and  .1 '  B 

=  100  ±  1  =  101  or  99.  When  the  auxiliary  curve  has  been  run, 
the  corresponding  stations  on  the  true  curve  are  found,  by  laying  off 
in  the  proper  direction  the  distances  B  B',  D  D',  &c.,  each  equal  to  b. 

77.  Proljlcm.  Having  run  a  curve  A  B  [Jig.  28),  to  change  the 
tangent  point  from  A  to  C,  in  such  a  way  that  a  curve  of  the  same  radius 
may  strike  a  given  point  D. 

Solution.  Measure  the  distance  B  D  from  the  curve  to  D  in  a  direction 
parallel  to  the  tangent  C E.  This  direction  may  be  sometimes  judged 
of  by  the  eye,  or  found  by  the  compass.  A  still  more  accurate  way  is 
to  make  the  angle  DBE  equal  to  the  intersection  angle  at  E,  or  to 
twice  BAE,  the  total  deflection  angle  from  A  to  B;  orif^  can  be 
seen  from  B,  the  angle  DBA  may  be  made  equal  to  BAE. 

Measure  on  the  tangent  (backward  or  forivard,  as  the  case  may  be)  a  dis 
lance  A  C  —  B  D,  and  C  will  be  the  7iew  tangent  point  required.     For.  if 
rfl"be  drawn  equal  and  parallel  to  A  F,  we  have  Fi7  equal  and  par 


MISCELLANEOUS    PROBLEMS.  5/ 

uUel  to  AC,  and  therefore  equal  and  parallel  to  B  D.  Hence  D  H  == 
B  F.=  AF=  CH,  and  D  /7  being  equal  to  C  H,  a.  curve  of  radios 
07  i^  from  the  tangent  point  C  must  pass  through  D. 


78  ProblenB.  Having  run  a  curve  A  B  (Jig.  29)  of  radius  li  <n 
deflection  angle  Z>,  terminating  in  a  tangent  B  D,  to  Jind  the  radius  IV  or 
deflection  angle  D'  of  a  curve  A  C,  that  shall  terminate  in  a  given  parallel 
tangent  CE. 


Fig.  29. 


A  K 

iSolution.     Since  the  radii  Z?  F  and  CG  are  perpendicular  to  the  par- 
allel tangents  CE  and  B  L>,  they  are  parallel,  and  the  angle  A  GG  = 
Therefore,  A  C  G,  the  half-supplement  of  A  G  C,  is  equal  t« 
4 


4.Fb 


m 


CIRCULAR    CURVES. 


A  B  F,  the  half-supplement  of  A  F  B.  Hence  A  B  and  B  C  are  in  the 
same  straight  line,  and  the  new  tangent  point  C  is  the  intersection  ol 
A  B  produced  with  C  E. 

Represent  AB  by  c,  and  A  C  =  c  -\-  B  C  by  c'.     Measure  B  C,  or,  if 
more  convenient,  measure  D  C  and  find  B  C  by  calculation.     To  calculate 

D  C 

B  C  from  D  C,  we  have  B  C  =^  ^-^^  j^^^  (Tab.  X.  9),  and  the  angle 

DBC  =  ABK=  BAK,  the  total  deflection  from  .4  to  B.     Then 

the  triangles  AFBandAG  C  give  A  B  :  AC  =  BF :  C  G,oy  c  :  c' 

=  R:R'; 

,'.R'  =  -R. 
c 


50 


50 


Sub- 


To  find  Z)',  we  have  (§  10)  /vl'  =  ^^^,  ,  and  R  =  ^^^  - 
sdtuting   these   values   in  the   equation   for   R',  we  have   gj^  jy,  = 


50 


TX 


50 
sin.  D  ' 


.  sin.  D'  =  -,  sin.  D. 


79.  Problem.      Given  the  length  of  tico  equal  chords  A  C  and  B  C 
[Jig.  30),  and  the  perpendicular  CD,  to  find  the  radius  R  of  the  curve. 


Fig.  30 


Solution.  From  0,  the  centre  of  the  curve,  draw  the  perpendicular 
OE.  Then  the  similar  triangles  QBE  and  BCD  give  B  0  :  B  E 
^  BC:  CD.orR:hBC=E  C:  CD.     Hence 


7?  = 


BC^ 
2  CD 


MISCELLANEOUS    PROBLEMS. 


59 


This  problem  serves  to  find  the  radius  of  a  curve  on  a  track  already 
laid.  For  if  from  any  point  C  on  tlie  curve  we  measure  two  equal 
.-hords  .1  Cand  B  C,  and  also  the  perpendicular  CD  from  Cu2)on  the 
whole  chord  A  B,  we  have  the  data  of  this  problem. 

80.  Prot>l.(3lll.  To  draw  a  tangent  F  G  {Ji<j.  30)  to  a  given  curve 
from  a  given  point  F. 

Solution.  On  any  straight  line  F/1,  ichich  cuts  the  curve  in  two  points, 
measure  F  C  arid  FA,  the  distances  to  the  curve.     Then,  by  Gcometrv, 


FG  =yFCx  FA. 

This  length  being  measured  from  F,  will  give  the  point  G.  When 
FG  exceeds  the  length  of  the  chain,  the  direction  in  which  to  measure 
it,  so  that  it  will  just  touch  the  curve,  may  be  found  by  one  or  two  trials. 

8\.  Problem.  Having  found  the  radius  A  0  ^  E  of  a  curve 
(fg.  31 ),  to  substitute  for  it  tico  radii  A  E  =  R^  and  D  F  =  A'o ,  (he, 
'ongcr  of  vhich  A  E  or  B  E '  is  to  be  used  for  a  certain  distance  only  ai 
mrh  end  of  the  curve. 


>Jolution.     Assume  the  longer  radius  of  any  length  ivhich  mat/  be  thought 


60  CIRCULAR    CURVES. 

proper,  and  find  (§  9)  the  corresponding  deflection  angle  D^.  Suppose 
that  each  of  the  curves  A  D  and  B  D'  is  100  feet  long.  Then  drawing 
CO,  we  have,  in  the  triangle  FOE,OE:FE  =  s'm.OFE :  sin.  FOE. 
But  the  side  OE  =  AE— AO  =  Ri  — R,  F E  =  D E  —  D F  == 
Z?i  —  /?<. ,  the  angle  FOE  =  \S0°  —  A  0  C  ^  1 80°  —  i  /,  and  the 
angle  0FE=A0F—  0EF=^I-2Di,  since  0  E  F  =  2  D, 
(§  7).  Substituting  these  values,  and  recollecting  that  sin.  (180°  —  ^7) 
=  sin.  ^  /,  we  have  R^  —  R\R^  —  R.  =  sin.  (i  /  —  2  Z), )  :  sin.  ^  1 
Hence 

'        sin.(i7-2Z)J 

^2  is  then  easily  found,  and  this  will  be  the  radius  from  D  to  D\  or 
until  the  central  angle  DFD'  =  I—  4  D^. 

The  object  of  this  problem  is  to  furnish  a  method  of  flattening  the 
extremities  of  a  sharp  curve.  It  is  not  necessary  that  the  first  curve 
should  be  ju'st  100  feet  long  ;  in  a  long  curve  it  may  be  longer,  and  in 
a  short  curve  shorter.  The  value  of  the  an^le  at  E  will  of  course 
change  with  the  length  of  A  D,  and  this  angle  must  take  the  place  of 
2  Di  in  the  formula.  The  longer  the  first  curve  is  made,  the  shorter 
the  second  radius  will  be.  It  must  also  be  borne  in  mind,  in  choosing 
the  first  radius,  that  the  longer  the  first  radius  is  taken,  the  shorter  will 
be  the  second  radius. 

Example.  Given  R  —  1146. 28  and  7=  45°,  to  find  i?2>  if  ^i  is  as- 
sumed =  1910.08,  and  A  D  and  B  D>  each  100.  Here,  by  Table  I., 
Dj  =  1°  30'.    Then 

A',  —R  =  763.8  2.8829S0 

i  /  =  22°  30'  sin.  9..582840 


2.465820 
i/— 2D,-=  19°  30'  sin.  9.523495 

Ri  —  R^  =  875.64  2.942325 

.-.  /?2  =  72i  —  875.64  =  1034.44 

82.  Problem.  To  locate  the  second  brcrch  of  a  compound  or  re- 
versed curve  from  a  station  on  the  first  branch. 

Solution.  Let  J.  B  (fig  32)  be  the  first  branch  of  a  compound  curve^ 
and  D  its  deflection  angle,  and  let  it  be  required  to  locate  the  second 
branch  AB\  whose  deflection  angle  is  Z)',  from  some  station  B 
unA  B. 


MISCELLANEOUS    PROBLEMS.  61 

Let  n  be  tfie  number  of  stations  from  A  to  B,  and  n'  the  number  of  sta- 
lions  from  A  to  any  station  B'  on  the  second  branch.  Represent  by  Vtht 
%ngle  A  B  B',  which  it  is  necessary  to  lay  off  from  the  chord  B  A  to  strike 
B>.    Let  the  correspondinj:;  ande  A  B'  B  on  the  other  curve  be  repre- 


Fig.  32 


rented  by  V.  Then  we  have  F+  F'  =  180°  —  BAB'.  But  if 
T T'  be  the  common  tangent  at  A,  we  have  TA  B  +  T'  A  B'  =  nD 
J^  n'  D'  =  180°  —  BAB'.  Therefore,  V-{-  V  =  nD  -{•  n' D'. 
Next  in  the  triangle  AB B'  we  have  sin.  V  :  sin.  V=  AB  :  AB'. 
But  A  B  :  A  B'  =  n  :n',  nearly,  and  sin.  V  :  sin.  V  =  V  :  V,  near- 

n 

ly.     Therefore  we  have  approximately  F' :  F  =  n  :  n',  or  F'  =  -,  F. 

Substituting  this  value  of  F'  in  the  equation  for  F+  F',  we  have 
r+ J  V=nD-\-n'D'.    Therefore,  n' F+ n  F=  ?i' (nZ)  +  n'Z)'),  or 

n  -\-  n' 

The  same  reasoning  will  apply  to  reversed  curves,  the  only  change 
being  that  in  this  case  F+  V  =  nD  —  n' D',  and  consequently 

V=  ^'  i»^  —  ^'D') 

n  -{•  n' 

When  in  this  formula  n' D'  becomes  greater  than  n  D,  V  becomes 
minus,  which  signifies  that  the  angle  Fis  to  be  laid  off  above  B  A  in- 
stead of  belov/. 

This  problem  is  particularly  useful,  when  the  tangent  point  of  a 
curve  is  so  situated,  that  the  instrument  cannot  be  set  o\cr  it.  The 
same  method  is  applicable,  when  the  curve  A  B'  starts  from  a  straight 
line ;  for  then  we  may  consider  A  B'  as  the  second  branch  of  a  com- 
pound curve,  of  which  the  straight  line  is  the  first  branch,  having  its 
radius  equal  to  infinity,  and  its  deflection  angle  D  =  0.  Making 
D  =  0,  the  formula  for  F  becomes 


62 


CIRCULAR    CURVES. 


n  -\-  )i' 

When  n  and  71'  are  each  1,  the  formula  for  Fis  in  all  cases  exact, 
for  then  the  supposition  that  V  :  V  =  71  :  n'  is  strictly  true,  since  AB 
will  equal  A  B',  and  Fand  F',  being  angles  at  the  base  of  an  isosceles 
triangle,  will  also  be  equal.     Making  n  and  71'  equal  to  1,  we  have 

When  the  curve  starts  from  a  sti-aight  line,  this  formula  becomes,  by 
making  Z)  =  0, 

We  have  seen  that  when  n  or  n'  is  more  than  1,  the  value  of  Fis 
only  approximate.  It  is,  however,  so  near  the  truth,  that  when  nei- 
ther n  nor  n'  exceeds  3,  the  error  in  curves  up  to  5°  or  6°  varies  from 
a  fraction  of  a  second  to  less  than  half  a  minute.  The  exact  value  of 
F  might  of  course  be  obtained  by  solving  the  triangle  ABB',  in 
which  the  sides  AB  and  AB'  may  be  found  from  Table  II.,  and  the 
included  angle  at  A  is  known.  The  extent  to  which  these  formnlte 
may  be  safely  used  may  be  seen  by  the  following  table,  which  gives 
the  approximate  values  of  Ffor  several  different  values  of  n,n',D^ 
and  />',  and  also  the  error  in  each  case. 


Compound  Curves. 

Reversed  Curves. 

n. 

D. 

0 

n". 

D'. 

0 

V. 

Error. 

n. 

D. 

0 

«'. 

0 

V. 

Error. 

0      ; 

i\ 

0     1 

n 

1 

0 

5 

1 

4  10 

0.9 

1 

3 

4 

3 

7    12 

27.2 

1 

0 

5 

3 

12  30 

25.3 

2 

3 

4 

3 

4      0 

23.5 

2 

0 

3 

3 

5  24 

22.1 

3 

3 

4 

3 

1    42f 

8.3 

3 

0 

3 

3 

4  30 

29.7 

3 

h 

0 

3 

3   45 

24.0 

1 

1 

5 

3 

13   20 

18.6 

2 

I 

1 

4 

0  40 

O.I 

2 

1 

2 
9 

1 

3 

1    20 

0.7 

2 

1 

4 

9 

4      0 

11.0 

2 

3- 

3 

7  48 

15.0 

1 

6 

2 

6 

4      0 

23.5 

0 

2 

4 

3 

10  40 

24.7 

1 

5 

3 

5 

7   .'U) 

51.8 

3 

3 

3 

4 

10  30 

54.0 

2 

3 

5 

3 

0  25f 

52.8 

As  the  given  quantities  are  here  arranged,  the  approximate  values 
of  Fare  all  too  great ;  but  if  the  columns  n  and  n'  and  the  columns  D 
and  D' were  interchanged,  and  F calculated,  the  approximntc  values 
of  F  would  be  just  as  much  too  small,  the  column  of  cnoi>  rcniaiuing 
the  same. 


MISCELLANEOUS    PROBLEMS. 


63 


83.  Problem.     To  measure  the  distance  across  a  river  on  a  given 
Uraight  line. 


D 


Fig.  3.3. 


Solution.  First  Method.  Let  A  B  (fig.  33)  be  the  required  distance 
Measure  a  line  A  C  along  the  bank,  and  take  the  angles  B  A  C  and 
ACB.     Then  in  the  triangle  ^1  C  Cwe  have  one  side  and  two  angles 

to  nnd  A  B. 

1(  A  Cis  of  such  a  length  that  an  angle  A  C B  =  ^D  A  C  can  he 
laid  off  to  a  point  on  the  farther  side,  we  have  ABC=^DAC=^ 
ACB.    Therefore,  without  calculation,  AB  =  AC. 


Fig.  34. 

Second  Method.  Lay  off  ^  C  (fig.  34)  perpendicular  to  A  B.  Meas- 
ure xi  C,  and  at  Clay  off  CZ)  perpendicular  to  the  direction  CB,  and 
meeting  the  line  of  /I  B  in  D.  Measure  A  D.  Then  the  triangles 
A  CD  and  ABC  are  similar,  and  give  AD  :  A  C  =-  A  C  :  AB. 

Therefore,  AB  ^  -^  . 

If  from  C,  determined  as  before,  the  angle  A  C  B'  be  laid  off  equal 
to  yl  CB,  we  have,  without  calculation,  A  B  =  AB'. 

Third  Method.  Measure  a  line  A  D  (fig.  35)  in  an  oblique  direction 
from  the  bank,  and  fix  its  middle  point  C  From  any  convenient 
point  E  in  the  line  of  A  B,  measure  the  distance  E  C,  and  prodiue 


64 


MISCELLANEOUS    PR0BLE3IS. 


E  C  until  CF=  Ea     Then,  since  the  triangles  A  CE  and  D  CF 
are  similar  by  construction,  we  see  that  DF  is  parallel  to  E  B.    Find 


Fig.  35 


now  a  point  G,  that  shall  be  at  the  same  time  in  the  line  of  CB  and 
of  D  F,  and  measure  G  D.  Then  the  triangles  ABC  and  D  G  C  sre 
equal,  and  G  D  is  equal  to  the  required  distance  A  B. 

As  the  object  of  drawing  E  Fis  to  obtain  a  line  parallel  to  A  B,  this 
line  may  be  dispensed  with,  if  by  any  other  means  a  line  GFhe  drawn 
through  D  parallel  to  AB.  A  point  G  being  found  on  this  parallel  in 
the  line  of  C B,  we  have,  as  before,  GD  =  AB. 


PARABOLIC    CURVES. 


65 


CHAPTER   II. 


PARABOLIC   CURVES. 
Article  I.  —  Locating  Parabolic  Curves. 

84.  Let  AEB  (fig.  36)  be  a  parabola,  A  C and  B  C  its  tangents, 
iiid  .1  B  the  chord  uniting  the  tangent  points.  Bisect  A  B  in  D,  and 
oin  CD.     Then,  according  to  Analytical  Geometry,  — 


Fig.  36. 


L    CD  is  a  diameter  of  the  parabola,  and  the  curve  bisects  CDinE- 

II.  If  from  any  points  T,  T',  T",  &c.,  on  a  tangent  A  F,  lines  be 
a.-awn  to  the  curve  parallel  to  the  diameter,  these  lines  T  M,  T'  M  , 
1  "M"  &c.,  called  tangent  deflections,  will  be  to  each  other  as  the 
Benares'  of  the  distances  AT,  A  T>,  A  T'\  &c.  from   the   tangent 

ptint  A. 

III.  A  line  F  D  (fig.  37),  drawn  from  the  middle  of  a  chord  A  Bio 
the  curve,  and  parallel  to  the  diameter,  may  be  called  the  middle  ordi 
nate  of  that  chord  ;  and  if  the  secondary  chords  A  E  and  B  E  he  drawn, 
the  middle  ordinates  of  these  chords,  K  G  and  /.  H.  are  each  equal  to 
{ED.  In  like  manner,  if  the  chords  A  A',  KE,EL,  and  LB  he 
drawn,  their  middle  ordinates  will  be  equal  to  \KG  or  \L  H. 

\V.  K  tangent  to  the  curve  at  the  extremity  of  a  middle  ordinate, 
is  parallel  to  the  chord  of  that  ordinate.  Thus  MF,  tangent  to  the 
cur\  e  at  E,  is  parallel  to  A  B. 


rs 


PARABOLIC    CURVES. 


V.  If  any  two  tangents,  as  yl  C  and  B  C,  be  bisected  in  M  and  / 
ihe  line  il/F,  joining  the  points  of  bisection,  will  be  a  new  tangent,  ita 
middle  point  E  being  the  point  of  tangency. 

85.  I*rol>leill.  Given  the  tangents  A  C  and  B  C,  equal  or  unequal^ 
{Jig.  36,)  and  the  chord  A  B,  to  lay  out  a  parabola  hy  tangent  deflections. 


Fig.  36 


Soluticm.  Bisect  A  B  in  A  and  measure  CD  and  the  angle  A  CD^ 
or  calculate  CD*  and  A  CD  from  the  original  data.  Divide  the  tan- 
gent A  C  into  any  number  n  of  equal  parts,  and  call  the  deflection 
JM/for  the  first  point  a.  Then  {§  84,  II.)  the  deflection  for  the  sec- 
ond point  will  be  T'  M'  =  4  a,  for  the  third  point  T"  M"  =  9  a,  and 
60  on  to  the  nth  point  or  C,  where  it  will  be  n^a.  But  the  deflection 
at  this  last  point  \sGE  =  ^CD{^  84,  I).  Therefore,  n^  a  =  C  E. 
and 

CE 


a  = 


n* 


Having  thus  found  a,  we  have  also  the  succeeding  deflections  4  a,  9  a. 
16  a,  &c.  Then  laying  ofl^  at  T,  T',  &c.  the  angles  A  T M,  A  T'  M>, 
&c.  each  equal  to  A  CD,  and  measuring  down  the  proper  deflections, 
just  found,  the  points  M,  il/',  &c.  of  the  curve  will  be  determined. 

The  curve  may  be  finished  by  laying  off  on  -4  C  produced  n  parts 
equal  to  those  on  A  C,  and  the  proper  deflections  will  be,  as  before,  a 
multiplied  by  the  square  of  the  number  of  parts  from  A.     But  an 


*  Since  C  D  is  drawn  to  the  middle  of  the  base  of  the  triangle  ^  iS  C,  we  have,  hj 
Rwmetrj-,  C  D'^  =  ^  (A  C^  +  B  C^)  —  A  D"-. 


LOCATING    PARABOLIC    CURVES. 


67 


PaMcr  way  generally  of  finding  points  beyond  E  is  to  divide  the  sec- 
ond tangent  B  Cinto  equal  parts,  and  proceed  as  in  the  case  of  ^  I. 
If  the  number  of  parts  on  B  C  be  made  the  same  as  on  A  (7,  it  is  obvi- 
ous that  the  deflections  from  both  tangents  will  be  of  the  same  length 
for  corresponding  points.     The  angles  to  be  laid  off  from  B  C  must, 

Df  course,  be  equal  to  BCD. 

The  points  or  stations  thus  found,  though  corresponding  to  equal 
distances  on  the  tangents,  are  not  themselves  equidistant.  The  length 
of  the  curve  is  obtained  by  actual  measurement. 

86.  Problem.  Given  the  tangents  A  C  and  B  C,  equal  or  unequal, 
[fig.  37,)  and  the  chord  A  B,  to  lay  out  a  parabola  by  middle  ordinates. 


Solution.  Bisect  A  B  in  D,  draw  CD,  and  its  middle  point  E  will 
oe  a  point  on  the  curve  (§  84,  L).  D  E  is  the  first  middle  oi^.nate, 
and  its  length  may  be  measured  or  calculated.  To  the  point  E  draw 
t>-.e  chords  A  E  and  BE,  lay  off  the  second  middle  ordinates  G  K  and 
HL,  each  equal  to  \DE{^  84,  III),  and  K  and  L  are  points  on  the 
curve.  Draw  the  chords  A  K,  K  E,  E  L,  and  L  B,  and  lay  oft  third 
middle  ordinates,  each  equal  to  one  fourth  the  second  middle  ordi- 
nates, and  four  additional  points  on  the  curve  will  be  determined. 
Continue  this  process,  until  a  sufficient  number  of  points  is  obtained 

87.  Prol>leiIl.      To  draiv  a  tangent  to  a  parabola  at  any  station. 

Solution.  I.  If  the  curve  has  been  laid  out  by  tangent  deflections 
(^  85).  let  M"'  (fig.  36)  be  the  station,  at  which  the  tangent  is  to  be 
drawn.  From  the'' preceding  or  succeeding  station,  lay  off,  parallel  to 
CD,  a  distance  M"NoxEL  equal  to  a,  the  first  tangent  deflection 
(§  85),  and  M'"  N  or  M'"  L  will  be  the  required  tangent.  The  same 
thing  may  be  done  by  laying  off  from  the  second  station  a  distance 
j^,  7^/  ^  4  „  or  at  the  third  station  a  distance  GP  =  ^a;   for  the 


BS  PARABOLIC    CURVES. 

required  tangent  will  then  pass  through  T'  or  G.  It  will  be  seen, 
also,  that  the  tangent  at  M'"  passes  through  a  point  on  the  tangent  at 
A  corresponding  to  half  the  number  of  stations  from  A  to  31'" ;  that 
is,  M'"  is  four  stations  from  A,  and  the  tangent  passes  through  T', 
the  second  point  on  the  tangent  A  C.  In  like  manner,  M'"  is  six  sta- 
tions from  Z?,  and  the  tangent  passes  through  G,  the  third  point  on  the 
tangent  B  C 

II.  If  the  curve  has  been  laid  out  by  middle  ordinates  (§  86),  the  tan- 
gent deflection  for  one  station  is  equal  to  the  last  middle  ordinate  made 
use  of  in  laying  out  the  curve.  For  if  the  tangent  A  C  (fig.  37)  were 
divided  into  four  equal  parts  corresponding  to  the  number  of  stations 
from  A  to  E^  the  method  of  tangent  deflections  would  give  the  same 
points  on  the  curve,  as  were  obtained  by  the  method  of  §  86.  In  this 
case,  the  tangent  deflection  for  one  station  would  be  a  =^  i\  C  E  ^ 
jg  DE.,  but  the  last  middle  ordinate  was  made  equal  to  ^  G  K  or 
ie  D  E.  Therefore,  a  is  equal  to  the  last  middle  ordinate,  and  a  tan- 
gent may  be  drawn  at  any  station  by  the  first  method  of  this  section. 

A  tangent  may  also  be  drawn  at  the  extremity  of  any  middle  ordi- 
nate, by  drawing  a  line  through  this  extremity,  parallel  to  the  chord 
of  that  ordinate  (§  84,  IV.). 

88.  In  laying  out  a  parabola  by  the  method  in  §  85,  it  may  some- 
times be  impossible  or  inconvenient  to  lay  off  all  the  points  from  the 
original  tangents.  A  new  tangent  may  then  bo  drawn  by  §  87  to  any 
station  already  found,  as  at  M'"  (fig.  36),  and  the  tangent  deflections 
a,  4  a,  9  a,  &c.  may  be  laid  off  from  this  tangent,  precisely  as  from  the 
first  tangent.  These  deflections  must  be  parallel  to  CD,  and  the  dis- 
tances on  the  new  tangent  must  be  equal  to  7'' iV  or  iViV",  which 
may  be  measured. 

89.  Problem.  Giveii  the  tangents  A  C  and  B  C,  equal  or  uneqiml, 
[Jiy  38,)  to  lay  out  a  parabola  by  bisecting  tangents. 

Solution.  Bisect  A  C  and  B  C  in  D  and  F,  join  D  F,  and  find  £",  the 
middle  point  of  D  F.  E  will  be  a  point  on  the  curve  (§  84,  V.).  We 
have  now  two  pairs  of  what  may  be  called  second  tangents,  A  D  and 
I)  E,  and  E  F  and  F  B.  Bisect  A  Din  G  and  D  E  in  H,  join  G  H, 
and  its  middle  point  ilf  will  be  a  point  on  the  curve.  Bisect  £"  F  and 
F  Bin  K  and  L,  join  KL,  and  its  middle  point  iVwill  be  a  point  on 
the  curve.  We  have  now  four  pairs  of  third  tangents,  A  G  and  G  M, 
M  H  and  U  E,  E  K  and  KN,  and  N L  and  L  B.  Bisect  each  pair  in 
turn,  join  the  points  of  bisection,  and  the  middle  points  of  the  joininj; 


LOCATING    PARABOLIC    CURVES. 


69 


lines  will  be  four  new  points,  il/',  M",  iV",  and  N'.   The  same  methcx? 
may  be  continued,  until  a  sufficient  number  of  points  is  obtained. 


Fig.  38. 


90.  Problem.      Given  the  tangents  A  C  and  B  C,  equal  or  unequal 
Hg.  39,)  and  the  chord  A  B,  to  lay  out  a  parabola  by  intersections. 


Fig.  39 


Solution.  Bisect  A  B  in  D,  draw  CD,  and  bisect  it  in  E.  Divide 
the  tangents  A  Cand  B  C,  the  half-chords  A  D  and  D  B,  and  the  line 
CE,  into  the  same  number  of  equal  parts  ;  five,  for  example.  Then 
the  intersection  M  of  A  a  and  F  G  will  be  a  point  on  the  curve.  For 
FM  =  I  Ca,  and  Ca  =  i  CE.  Therefore.  FM=  55  CE,  which  is 
the  proper  deflection  from  the  tangent  atFto  the  curve  (§  8.5).  In 
like  manner,  the  intersection  N  of  Ab  and  II K  may  be  shown  to  be  a 
point  on  the  curve,  and  the  same  is  true  of  all  the  similar  intersections 
indicated  in  the  figure. 

If  the  line  DE  were  also  divided  into  five  equal  parts,  the  line  A  a 
would  be  intersected  in  il/on  the  curve  by  a  line  drawn  from  B  through 
a',  the  line  A  b  would  be  intersected  in  iVon  the  cur\'e  by  a  line  drawn 


70 


PARABOLIC    CURVES. 


from  B  through  6',  and  in  general  any  two  lines,  drawn  from  A  and  B 
through  two  points  on  CD  equally  distant  from  the  extremities  Cand 
D,  will  intei-sect  on  the  curve.  To  show  this  for  any  point,  as  x)/,  it  is 
sufficient  to  show,  that  B  a'  produced  cuts  F  G  on  the  curve  ;  for  it 
has  already  been  proved,  that  A  a  cuts  F  G  on  the  curve.  Now 
Da':MG^BD:B  G  =  b:^,or  M  G  =lDaK  But  Da' =  \  C  E. 
Therefore,  MG  =  h  C E.  Again,  F  G  :  CD  =^  A  G  :  A  D  =  I  ■:>. 
Therefore,  FG  =  \CD  =  lCE.  We  have  then  FM  =  F  G  — 
MG  =  f  CE  —  ii  C E  =  is  C E.  As  this  is  the  proper  deflection 
from  the  tangent  at  F  to  the  cm-ve  (§  85),  the  intersection  of  B  a'  with 
F  G  is  on  the  curve.  This  furnishes  another  method  of  laying  out  a 
parabola  by  intersections. 

91.  The  following  example  is  given  in  illustration  of  several  of  the 
preceding  methods. 

Example.  Given  AC  =  B  C  ^  832  (fig.  40),  and  -1  B  =  1536  to 
lay  out  a  parabola  A  E B.  We  here  find  CD  =  320.  To  begin  with 
the  method  by  tangent  deflections  (§  85),  divide  the  tangent  A  C  into 

C  E         ^(\0 

eight  equal  parts.     Then  a  =  —^  =  -wr  =  2.5.     Lay  off  from  the 

divisions  on  the  tangent  Fl  =  2.5,  G2  =4  X  25  =  10,  ^3  = 
9X25  =  22.5,  and  /v  4  =  16  X  2.5  =  40.  Suppose  now  that  it  is 
inconvenient  to  continue  this  method  beyond  K.    In  this  case  we  may 


Fig.  40 


find  a  new  tangent  at  E,  by  bisecting  A  Cand  B  C  {^  89),  and  draw- 
ing KL  through  the  points  of  bisection.  Divide  the  new  tangent 
KE  =^  ^  AD  ^  384  into  four  equal  parts,  and  lay  oflT  from  KE  the 


RADIUS    OF    CURVATURE. 


71 


same  tangent  deflections  as  were  laid  off  from  .fi  iiT,  namely,  3/5 - 
22.5  A^6  =  10,  and  07  =  2.5.  To  lay  off  the  second  half  of  the 
curve  by  middle  ordinates  (§86),  measure  EB=  784.49.  Bisect 
EB  in  P,  and  lay  off  the  middle  ordinate  P  R  =  ^D  E  ^  AQ. 
Measure  ER^  386.08,  and  BR  =  402.31,  and  lay  off  the  middle  or- 
dinates S  T  and  V  IF,  each  equal  to  ^  P  /2  =  10.  By  measuring  the 
chords  ET,  TR,  R  TF,  and  WB,  and  laying  off  an  ordinate  fron' 
each,  equal  to  2  5.  four  additional  points  might  be  found. 


Article  II. —  Radius  of  Curvature. 

92.  The  curvature  of  circular  arcs  is  always  the  same  for  the  same 
arc,  and  in  different  arcs  varies  inversely  as  the  radii  of  the  arcs. 
Thus,  the  curvature  of  an  arc  of  1,000  feet  radius  is  double  that  of  an 
arc  of  2,000  feet  radius.  The  curvature  of  a  parabola  is  continually 
changing.  In  fig.  39,  for  example,  it  is  least  at  the  tangent  point  A, 
the  extremity  of  the  longest  tangent,  and  increases  by  a  fixed  laAv,  un- 
til it  becomes  greatest  at  a  point,  called  the  vertex,  where  a  tangent  to 
the  curve  would  be  perpendicular  to  the  diameter.  From  this  poin; 
to  B  it  decreases  again  by  the  same  law.  We  may,  therefore,  con- 
sider a  parabola  to  be  made  up  of  a  succession  of  infinitely  small  cir- 
cular arcs,  the  radii  of  which  continually  increase  in  going  from  the 
vertex  to  the  extremities.  The  radius  of  the  circular  arc,  correspond- 
ing to  any  part  of  a  parabola,  is  called  the  radius  of  curvature  at  that 

point. 

If  a  parabola  forms  part  of  the  line  of  a  railroad,  it  will  be  necessa- 
ry, in  order  that  the  rails  may  be  properly  curved  (§  28),  to  know 
how  the  radius  of  curvature  may  be  found.  It  will,  in  general,  be 
necessary  to  find  the  radius  of  curvature  at  a  few  points  only.  In 
short  curves  it  may  be  found  at  the  two  tangent  points  and  at  the  mid- 
dle station,  and  in^onger  curves  at  two  or  more  intermediate  points 
besides.  The  rails  curved  according  to  the  radius  at  any  point  should 
be  sufficient  in  number  to  reach,  on  each  side  of  that  point,  half-way  to 
the  next  point. 

93.  Problem.     To  find  the  radius  of  curvature  at  certain  stations 

on  a  parabola. 

Solution.    Let  AEB  (fig.  41)  be  any  parabola,  and  let  it  be  re- 
quired to  find  the  radii  of  curvature  at  a  certain  number  of  stations 


72 


PARABOLIC    CURVES. 


fron.  A  to  E.     Tliese   stations  must  be  selected  at  regular  interral 
from  those  determined  by  any  of  the  preceding  methods.     Let  n  de 
note  the  number  of  parts  into  which  ^  £  is  divided,  and  divide  CL 
into  the  same  number  of  equal  parts.    Draw  lines  from  A  to  the  points 


of  division.     Thus,  if  n  —  4,  as  in  the  figure,  divide  CD  into  four 
equal  parts,  and  draw  A  F,  A  E,  and  A  G.     Let  A  D  =  c^  A  F  =  Ci 
A  E  =  C2,  A  G  —  C3,  and  A  C  =  T.     Denote,  moreover,  C  D  hy  d 
and  the  area  of  the  triangle  A  C B  hy  A.     Then  the  respective  radii 
for  the  points  .£,1,2,  3,  and  A  will  be 


R  =  2,      /?,  = 


A 


II 


V2 


A 


A*3 


A  ' 


Ra  = 


A 


The  area  A  may  be  found  by  form.  18,  Tab.  X.;  c  and  T  are  known ; 
and  Ci,  Co,  c^  may  be  found  approximately  by  measurement  on  a  figure 
carefully  constructed,  or  exactly  by  these  general  formulae  :  — 


&c. 


7^2  _c2        {n~\)d^ 


n 

f2 

— 

c2 

n 

j'i 

— 

C2 

n 

'fi 

— 

c2 

n' 

[n 

-3) 

d^ 

n2 

[n 

-5) 

f/2 

n2 

[n_ 

-7) 

«2 

d^ 

&c. 


It  will  be  seen,  that  each  of  these  values  is  formed  from  the  preceding, 
by  adding  the  same  quantity  — - — ,  and  subtracting  ^  multiphed  in 
STiccess-lorj  hr  w  —  1,  n  -    Z  -n   -    5,  v^     ^flaking:  ^>  ~  i,  we  have 


RADIUS    OF    CURVATUKb. 


ra 


c^^  =  c^  4-^(r2_c«)-i'gcr', 


C2''  =  c,^-hUT'-c-')-ud\ 


Ca^  =  c 


i' -\-  ^  {T^  -  c'')  +  ud'. 


A.11  the  quantities,  wliicb  enter  in*  j  tlic  expressions  for  the  radii,  are 
now  known,  and  the  radii  may,  therefore,  be  determined.  The  same 
method  will  apply  to  the  other  half  of  the  parabola. 

The  manner  of  obtaining  the  preceding  formulte  is  as  follows.  The 
radius  of  curvature  at  any  given  point  on  a  parabola  is,  by  the  Differ- 
ential Calculus.  R  =  2^i^^.3  E  '  ^"  which  p  represents  the  parameter  of 
I  lie  parabola  for  rectangular  coordinates,  and  E  the  angle  made  with 
a  diameter  by  a  tangent  to  the  curve  at  the  given  point.  First,  let  the 
middle  station  E  (fig.  42)  be  the  given  point.    Then  the  angle  E  is  the 


Fig.  42 


angle  made  with  E  Dhy  n  tangent  at  E,  or  since  A  B  is  parallel  to 
the  tangent  at  E  (§  84,  IV.),  sin.  E  =  sin.  ADE  =  sin.  BDE.  Let 
p'  be  the  parameter  for  the  diameter  E  D.     Then,  by  Analytical  Ge 


ometry,  f 

p'  8in.2  E 
2  8in.3  E  ^ 
c3 


p'   sin  2  E.      Therefore,  at  this   point  R  = 


2  8in.3  E  ~ 
2sihE ■    ^^^ P'-^^  =  Vd'    Therefore,  R  =  j^ 

--=     .  .         =  —  :  since  A  ^^  cd  sin.  E  (Tab.  X.  17). 
c  d  sin.  E         A  ^  ^  ' 

Next,  to  find  7?i ,  or  the  radius  of  curvature  at  H,  the  first  station 
from  E.  Through  ff  draw  EG  parallel  to  CD,  and  from  Fdraw  the 
tangent  EK.  Join  A  K,  cutting  C  Dm  L.  Then  from  what  has  just 
been  pioved  for  the  radius  of  curvature  at  E,  we  have  for  the  radius 

of  curvature  at  //.  A',  =   a  F  K'      ^^^^  A  G  •  A  L  =  A  F :  A  C  = 


74 


PARABOLIC    CURVES. 


n~  I  :  n,  or  A  G  =  -  ~  x  A  L.  But  A  L  =  c,  For,  Miice  A  F  - 
—^  X  AC,  the  tangent  deflection  FH  =  ^"  ~/^"  .  ^  (§  84,  II.),  and 
FG  =  2FH=^-^^^^d.     Then,  since  CL:FG  =  AC:AF  = 

n:n-l,CL  =  ^^X  FG='^d.  Hence  L  D  =  d  - '^  d 
=  -  c7,  thut  is,  .1  L  =  Ci  .  Substituting  this  value  in  the  expres- 
sion  for  A  G  above,  we  have  A  G  =  -^—  c^  .  Moreover,  since 
A  F  =  — - —  X  A  C,  a/id  because  similar  triangles  are  to  each  other  a? 
the  squares  of  their  homologous  sides,  we  have  the  triangle  A  F  G  = 
^"  ~  ^^'  X  A  CL.  But  ACL:ACD=^CL:CD  =  n  —  l:  n,  or 
ACL=^  "^  X  A  CD.    Therefore,  A  F  G '=  ^-^^^~^  X  A  C D,  and 

AFK  =  2AFG  =  ^^^^^  XACB  =  ^^.'^  A.  Substituung 
these  values  of  A  G  and  A  F K  in  the  equation  R^  =  j^p^  ,  and  re- 
ducing,  we  find  7?j  =  —  .    By  similar  reasoning  we  should  find  /?2  = 

It  remains  to  find  the  values  of  Cj  ,  c. ,  &c.  Through  A  draw  J  ili 
pei-pendicular  to  CD,  produced  if  necessary.  Then,  by  Geometry,  we 
have  AD"^  =  A  L"  +  L  D"  —  2  L  D  X  LM,  and  AC  =  A  L^  -{- 
CU  +  2  CL  X  L  M.  Finding  from  each  of  these  equations  the 
value  of  2  L  M,  and  putting  tliese  values  equal  to  each  other,  we  have 

zT^ = CL •    ^"^  AL  =  Ci,LD=-d, 

n 1 

A  D  =  c,  A  C  =^  2\  and   CL  =  -^ —  d.      Substituting  these  values 

in  the  last  equation,  and  reducing,  we  find 


r^       (»  — l)c2        [n  —  \)d^ 


^^    -    »    +  n  ~         n^ 


By  similar  reasoning  we  should  find 


2  7^2        (u  —  2)c2       2{n  —  2)d 


« 


c,^=  -:r  + 


s 


n  n  n 


3  r«       (tt  — 3)c«       3(n  — airf" 


&c.  &c. 


RADIUS    OF    CURVATURE. 


75 


From  tlicsc  equations  the  values  of  c,S  Co',  Cj"^ ,  &c.  given  on  page  72 
arc  readily  obtained.  That  given  for  Cj'  is  obtained  from  the  first  ol 
these  equations  by  a  simple  reduction  ;  that  given  for  Cj-  is  obtained 
by  subtracting  the  first  of  these  equations  from  the  second,  and  reduc- 
ing ;  that  given  for  c^^  is  obtained  by  subtracting  the  second  equation 
from  the  third,  and  reducing ;  and  so  on. 

94.  Example.  Given  (fig.  A\)  A  C  ^  T  ^  600,  B  C  ==  T<  ^  520, 
and  AD  =  c  =  550,  to  find  R,  R^ ,  H, ,  R3 ,  and  R^ ,  the  radii  of  cur- 
vature at  .E,  1 ,  2,  3,  and  A. 

To  find  CD  =  d,  we  have,  by  Geometry,  d^=^[T-  -{-  7''  ^j  —  c« 

■ 

which  gives  d-  =  12700. 
To  find  the  area  of  .1  CB  =  A,  we  have   (Tab.  X.   18)   A  = 

./sis  —a)  (s  —6)  is—c) . 

*^     ^  '  s  =  1110  3.045323 

c  — a  =  590  2.770852 

s  —  6  =  510  2.707570 

s  —  c  =  10  1.000000 


2)9.523745 


lojr.  A  4.761872 


■'to 


Next  ^  (r^  -  0')  =  i  (r  +  c)  (r-  c)  =  ii5!^  =  14375,  and 
t  „lf  «L  =  793.75.     Then 

•*    lb 

c^    =  550-  =  302500 

Cj^  =  302500  +  14375  —  3  X  793.75  =  314493.75 

Co^  ==  314493.75  +  14375  —  793.75  =  328075 

C32  =  328075  +  14375  +  793.75  =-  343243.75 

C3 

To  find  /?,  we  have  /2  =  ^  ,  or  log.  R  =  3  log.  c  —  log.  A. 

c    =  550  2.740363 

c^  8.221089 

A  4.7618^ 

22  =  2878.8  3.459217 

To  find  Rj, ,  we  have  Ri  ==  ^  >  or  log.  Ri  =-2-log  Cj^  —  log.  A. 

Cj^  =  314493.75  5.49761 

c,3  8.246418 

A  4.76  872 

i?,  ==  3051.7  3.484546 


76  PARABOLIC    CURVES. 

In  the  same  way  we  should  find  i?2  =  3251.5,  R^  =  3479.6,  R^  ^ 
3737.5. 

To  find  the  radii  for  the  second  part  E  B  o(  the  parabola,  the  same 
formulse  applv,  except  that  T'  takes  the  place  of  T.     We  have  then 

l(r-  -  c',"=  UT'  +  c)  ,r  -  0  =  15™^^  =  _su.5 

Hence 

Ci*''  =  302.500  —  S025  —  2381.25  =  292093.75 
C2^  =  292093.75  —  8025  —  793.75  =  283275. 
C32  =  2S3275  —  8025  +  793.75  :=  276043.75 

C    3  3 

To  find  Ri ,  we  have  /?i  =  -y  ,  or  log.  Ri  =  5  log  Ci"-'  —  log.  A 

c  2  =  292093.75  5.465523 


c^  8.198284 

A  4.761872 


/?,  =  2731.6  3.436412 

In  the  same  way  we  should  find  R<i_  =  2608.8,  R^  =  2509.5,  R^  -=> 
2433. 

It  will  be  seen,  that  the  radii  in  this  example  decrease  from  one  tan- 
gent point  to  the  other,  which  shows  that  both  tangent  points  lie  on 
the  same  side  of  the  vertex  of  the  parabola  (§  92).  This  will  be  tho 
case,  whenever  the  angle  BCD,  adjacent  to  the  shorter  tangent,  ex- 
ceeds 90°,  that  is,  whenever  c'  exceeds  T'^  -\-  d}.  If  B  CD  =  90°. 
the  tangent  point  B  falls  on  the  vertex.  If  BCD  is  less  than  90°, 
one  tangent  point  falls  on  each  side  of  the  vertex,  and  the  curvature 
will,  therefore,  decrease  towards  both  extremities. 

95.  If  the  tangents  T  and  T'  are  equal,  the  equations  for  c,',  Co',  &c. 
will  be  more  simple;  for  in  this  case  d  is  perpendicular  to  c,  and  T' 
—  c^  =  d^.     Substituting  this  value,  we  get 

d^ 

3d^ 
Co   =  Cj   -4-  -^  , 

5d^ 

&C.  &C. 

example.     Given,  as  in  §  91,    T  ^  T'  =  832,  c  =  768,  and  d  = 


RADIUS    OF    CURVATURE. 


n 


320,  to  find  the  radii  /?,  Ri ,  and  R^  at  the  points  E,  4,  and  A  (fig.  40) 
Here  A  =  cd  =  245760,  n  =  2,  and  c,'  =  c^  +  |£/2  =  615424 

c3  c2         7G82  _  C,3 

Then  /?  ^.  ^ 


C2  7G82  C,3  .     _  r3 


^'  '''i  =  75  ' ''"" 

c,2  =  615424 

5.789174 

cd  =  245760 

8.683761 
5.390511 

/?!  =  :  964.5 
r=  832 

3.293250 
2.920123 

23 

erf  =  245760 

8.760369 
5.390511 

R..  =  2343.5 

3.369858 

W  is  the  radius  at  the  point  R  also,  and  7?,  the  radius  at  the  point  B 


78  LEVELLING. 


CHAPTER   IIL 

LEVELLING. 

Article  I. —  Heights  and  Slope  Stakes. 

96.  The  Level  is  an  instriiinent  consisting  essentially  of  a  telesco]>e. 
.supported  on  a  tripod  of  convenient  lieight,  and  capable  of  being  so 
adjusted,  that  its  line  of  sight  shall  be  horizontal,  and  that  the  tel- 
escope itself  may  be  turned  in  any  direction  on  a  vertical  axis.  The 
instrument  when  so  adjusted  is  said  to  be  set. 

The  line  of  sight,  being  a  line  of  indefinite  length,  may  be  made  to 
describe  a  horizontal  plane  of  indefinite  extent,  called  the  plane  of  Om 
lei-el. 

The  levelling  rod  is  used  for  measuring  the  vertical  distance  of  any 
point,  on  which  it  may  be  placed,  below  the  plane  of  the  level.  Thi? 
distance  is  called  the  sight  on  that  point. 

97.  Pro1>lcill.  To  Jind  the  difference  of  level  of  two  points,  as  A 
and  B  [fig.  43). 

Solution.  Set  the  level  between  the  two  points,*  and  take  sights  on 
both  points.  Subtract  tlie  less  of  these  siglits  from  the  greater,  and 
the  difference  will  be  the  difference  of  level  required.  For  i(  F  P  rep- 
resent the  plane  of  the  level,  and  A  G  he  drawn  through  ^4  parallel  to 
FP,  A  F  will  be  the  sight  on  A,  and  B  P  the  sight  on  B.  Tiien  the 
required  difference  of  level  B  G  =  BP  -   ^^G  =  BP  —  AF. 

If  the  distance  between  the  points,  or  rue  nature  of  the  ground, 
makes  it  necessary  to  set  the  level  more  than  once,  set  down  all  the 
backward  sights  in  one  column  and  all  the  forward  sights  in  another. 
Add  up  these  columns,  and  take  the  less  of  the  two  sums  from,  the 
greater,  and  the  difference  will  be  the  difference  of  level  required. 
Thus,  to  find  the  difference  of  level  between  A  and  D  (fig.  43),  the 
level  is  first  set  between  A  and  .B,  and  sights  are  taken  on  A  and  B ; 
the  level  is  then  set  between  B  and  C,  and  sights  are  taken  on  B  and 

*  The  level  should  be  placed  midway  between  the  two  points,  when  practicable, 
In  order  to  neutralize  the  effect  of  inaccuracy  in  the  adjustment  of  the  instrument, 
And  for  the  reason  given  in  §  i05. 


TIEir.IITS    AND    SLOPE    STAKES. 


79 


C,    lastly:  the  level  is  set 


pa     o 


usually  divided  into  regular 
the  datum  plane  is  required 


between  C  and  D,  and  sights  are  taken 
on  6' and  D.  Then  the  ditlcrence  of 
level  between  ^1  and  D  \s  E  D  = 
{BP+  KC-\-  OD)  —  [AF-VBJ-\- 
NC).  For  E  D  ==  no  -  LC  ^ 
tl M -\-MC—L  C.  llutllM  =  h  G 
=  BF-  AF,  MC  =^KC  -  D I, 
and  L  C  =^  N  C  —  0  D.  Sal)stituting 
these  values,  we  have  ED  =  BP  — 
AF-\-  KG  -BI  -  iVC+  0D  = 
(BP-]-  KG  +  CD)  —  {AF+  Bl 

-^  NC). 

98.  It  is  often  convenient  to  refer  all 
heights   to   an    imaginary   level    plane 
called    the    Jalum  plane.      This    plane 
may    be  assumed   at  starting   to   pass 
through,  or  at  some  fixed  distance  above 
or  below,  any  permanent  o1)ject,  called 
a  bmch-mark,  or  simply  a  bench.     It  is 
most  convenient,  in  order  to  avoid  mi- 
nus heights,  to  assume  the  datum  plane 
at  such  a  distance   below  the   bench- 
mark, that  it  will  pass  below  all  the 
points  on  the  line  to  be  levelled.    Thus 
if  A  F>  (tig.  44)  were  part  of  the  line  to 
be  levelled,  and  if  A  were  the  starting 
point,   we   should    assume   the   datum 
plane   GD  at   such  a  distance   below 
some    permanent    object    near   A,   as 
would  make  it  pass  below  all  the  points 
on  the  line.     If,  for  instance,  we  had 
reason  to  believe  that  no  point  on  this 
line  was  more  than  15  or  20  feet  below 
A,  we  might  safely  assume  G D  to  be 
25   feet  below  the   bench   near  A,  in 
which  case  all  the  distances  from  the 
line  to  the  datum  plane  would  be  posi- 
tive.    Lines  before  being  levelled  are 
stations,  the  height  of  each  of  which  above 


80 


LEVELLING. 


^9.  Prol>!eill.    To  find  the  heights  above  a  datum  plane  of  the  sev 

eral  stations  on  a  given  line. 

Solution.    JjetA  B  (fig.  44)  represent 
a  portion  of  the  line,  divided  into  regu 
lar  stations,  marked  0,  1,2,  3,  4,  5,  «Sbc 
and  let  CD  represent  the  datum  plane, 
assumed  to  be  25  feet  below  a  bench- 
mark near  .1.     Suppose  the  level  to  be 
set  first  between  stations  2  and  3,  and  a 
sight  upon  the  bench-mark  to  be  taken, 
and  found  to  be  3.125.     Now  as  this 
sight  shows  that  the  plane  of  the  level 
E  F'ls  3.125  feet  above  the  bench-mark 
and  as  the  datum  plane  is  25  feet  bo 
low  this  mark,  we  shall  find  the  height 
of  the  plane  of  the  level  above  the  da 
turn   plane   by   adding   these    heights, 
which  gives  for  the  height  of  E  F  25  -\- 
3.125  =  28.125  feet      This  height  mav 
for  brevity's  sake  be  called   the  height 
of  the  instrument,  meaning  by  this  the 
height  of  the  line  of  sight  of  the  instru 
ment. 

If  now  a  sight  be  taken  on  station  0, 
vcQ  shall  obtain  the  height  of  this  sta- 
tion above  the  datum  plane,  by  sub- 
tracting this  sight  from  the  height  of 
the  instrument ;  for  the  height  of  this 
station  is  0  C  and  OC=EC— EO. 
Thus  if  EO  =  3  413,  0  C  =  28.125  — 
3.413  =  24.712.  In  like  manner,  the 
heights  of  stations  1,  2,  3,  4,  and  5  may 
be  found,  by  taking  sights  on  them  in 
succession,  and  subtracting  these  sights 
from  the  height  of  the  instrument. 
Suppose  these  sights  to  be  respective- 
ly 3.102,  3.827,  4.816,  6.952,  and  9.016, 
and  we  have 
=  28.125  —  3.413  =  24.712, 


height  of  station  0 


1  =  28.125  —  3.102  =  25.023, 


HEIGHTS    AND    SLOPE    STAKES.  81 

height  of  station  2  =  28.125  —  3.827  =  24.298, 
"  "  "  3  :=  28.125—  4.816  =  23.309, 
''  "  "  4  =  28.125  —  6.952  =  21.173, 
«        *'      "         5  =  28.125  —  9.016  =  19.109. 

Next,  set  tlie  level  between  stations  7  and  8,  and  as  the  height  of  sta- 
tion 5  is  known,  take  a  sight  upon  thTs  point.  This  sight,  being  added 
to  the  height  of  station  5,  will  give  the  height  of  the  instrument  in  its 
new  position  ;  for  G  K  =  6'  5  +  5  K.  Suppose  this  sight  to  be  G  5 
=  2.740,  and  we  have  GK=  19.109  +  2.740  =  21.849.  A  point 
like  station  5,  which  is  used  to  get  the  height  of  the  instrument  after 
resetting,  is  called  a  turning  point.  The  height  of  the  instrument  being 
found,  sights  are  taken  on  stations  6,  7,  8,  9,  and  10,  and  the  heights 
of  these  stations  found  by  subtracting  these  sights  from  the  height  of 
the  instrument.  Suppose  these  sights  to  be  respectively  3.311,  4.027, 
3.824,  2.516,  and  0.314,  and  we  have 

height  of  station  6  =  21.849  —  3.311  =  18.538, 
■'  "  "  7:^-21.849  —  4.027  =  17.822, 
"  "  "  8  =  21.849  —  3.824  =  18.025, 
«  »  «  9  =  21.849  —  2.516  =  19.333, 
"        "      "       10  =  21.849  —  0.314  =  21.535. 

The  instrument  is  now  again  carried  forward  and  reset,  station  IC 
IS  used  as  a  turning  point  to  find  the  height  of  the  instrument,  and 
every  thing  proceeds  as  before. 

At  convenient  distances  along  the  line,  permanent  objects  are  se 
lected,  and  their  heights  obtained  and  preserved,  to  be  used  as  starting 
points  in  any  further  operations.  These  are  also  called  benches.  Let 
us  suppose,  that  a  bench  has  been  thus  selected  near  station  9,  and 
that  the  sight  upon  it  from  the  instrument,  when  set  between  stations 
7  and  8,  is  2.635.  Then  the  height  of  this  bench  will  be  21.849  — 
2.635  =  19  214. 

100.  From  what  has  been  shown  above,  it  appears  that  the  first 
thing  to  be  done,  after  setting  the  level,  is  to  take  a  sight  upon  some 
point  of  known  height,  and  that  this  sight  is  always  to  be  added  to  the 
known  height,  in  order  to  get  the  height  of  the  instrument.  This  first 
sight  may  therefore  be  called  a  p///s  sight.  The  next  thing  to  be  done 
is  to  take  sights  on  those  points  whose  heights  are  required,  and  to 
subtract  these  sights  from  the  height  of  the  instrument,  in  order  to  get 
the  required  heights.  These  last  sights  may  therefore  be  called  mimia 
sights 


82 


LEVELLING. 


101.  The  field  notes  are  kept  in  the  following  form.  The  first  col 
umn  in  the  table  contains  the  stations,  and  also  the  benches  marked 
B.,  and  the  turning  points  marked  t.  p.,  except  when  coincident  wuli 
a  station.  The  second  column  contains  the  plus  sights  ;  the  third  col- 
umn shows  the  height  of  the  instrument  ;  the  fi)urth  contains  the  ininus 
sights  ;  and  i\iQ  fifth  contains  the  heights  of  the  points  in  the  first  column. 


Station 

+  s. 

H.I. 

—  S. 

1 

n. 

B. 

3.125 

25.000 

0 

28.125 

3.413 

24.712 

1 

3.102 

25.023 

2 

3  827 

24.298 

3 

4.816 

23.309 

4 

6.952 

21.173 

5 

2.740 

9.016 

19.109 

6 

21.849 

3311 

18.538 

7 

4.027 

17.822 

8 

.   3.S24 

18.025 

9 

2.516 

19.333 

B. 

2.635 

19.214 

10 

0.314 

21.535 

The  height  of  the  bench  is  set  down  as  assumed  above,  namely,  25 
feet;  the  first  plus  sight  is  set  opposite  B.,  on  which  point  it  was 
taken,  and,  being  added  to  the  height  in  the  same  line,  gives  the  height 
of  the  instrument,  which  is  set  opposite  0 ;  the  minus  sights  are  set 
opposite  the  points  on  which  they  are  taken,  and,  being  subtracted 
from  the  height  of  the  instrument,  give  the  heights  of  these  points,  as 
set  down  in  the  fifth  column.  The  minus  sights  are  subtracted  from 
the  same  height  of  the  instrument,  as  far  as  the  turning  point  at  station 
5,  inclusive.  The  plus  sight  on  station  5  is  set  opposite  this  station, 
and  a  new  height  obtained  for  the  instrument  by  adding  the  plus  sight 
to  the  height  of  the  turning  point.  This  new  height  of  the  instrument 
is  set  opposite  station  6,  where  the  minus  sights  to  be  subtracted  from 
it  commence.  These  sights  are  again  set  opposite  the  points  on  which 
they  were  taken,  and,  being  subtracted  from  the  new  height  of  the  in- 
strument, give  the  heights  in  the  last  column. 

102.  Problem.  To  set  slope  stakes  for  excavations  and  embank- 
ments. 

Solution.  Let  A  B  H K  C  (fig.  45)  be  a  cross-section  of  a  proposed 
excavation,  and  let  the  centre  cut  A  M  =  c,  and  the  width  of  the  road 


HEIGHTS    j'ND    SLOPE    STAKES. 


83 


fM}d  II K  =  b.  The  slope  of  the  sides  B  H  or  C  Kis  usually  given  by 
the  ratio  of  the  base  K  Nto  the  height  E  N.  Suppose,  in  the  present 
case,  that  KN :  E  N  ^  3  :  2,  and  we -have  the  slope  =  I .  Then  if 
the  ground  were  level,  as  D  A  E,  it  is  evident  that  the  distance  from 


Fig.  45 


the  centre  A  to  the  slope  stakes  at  D  and  E  would  be  yl  Z)  =  A  E  — 
M  K -\-  KN=^b  +  I  c.  But  as  the  ground  rises  from  A  to  C 
t!i rough  a  height  C  G  =  g,  the  slope  stake  must  be  set  farther  out  a 
distance  E  G  =  ^  g ;  and  as  the  ground  falls  from  A  to  B  through  a 
height  B  F  =^  g,  the  slope  stake  must  be  set  farther  in  a  distance  D  F 

3 
=   2  9- 

To  find  B  and  C,  set  the  level,  if  possible,  in  a  convenient  position 
for  sighting  on  the  points  A^  B,  and  C.  From  the  known  cut  at  the 
centre  find  the  value  oi  AE  =  ^h  -{-^c.  Estimate  by  the  eye  the 
rise  from  the  centre  to  where  the  slope  stake  is  to  be  set,  and  take  this 
as  the  probable  value  of  g.  To  A  E  add  |  g,  as  thus  estimated,  and 
measure  from  the  centre  a  distance  out,  equal  to  the  sum.  Obtain 
now  by  the  level  the  rise  from  the  centre  to  this  point,  and  if  it  agrees 
with  the  estimated  rise,  the  distance  out  is  correct.  But  if  the  esti- 
mated rise  prove  too  great  or  too  small,  assume  a  nev.^  value  for  g, 
measure  a  corresponding  distance  out,  and  test  the  accuracy  of  the 
estimate  by  the  level,  as  before.  These  trials  must  be  continued,  until 
the  estitnated  rise  agrees  sufficiently  well  with  the  rise  found  by  the 
level  at  the  corresponding  distance  out.  The  distance  out  will  then  be 
hb  -\-  2*^  -{•%  g-  The  same  course  is  to  be  pursued,  when  the  ground 
falls  from  the  centre,  as  at  Z? ;  but  as  g  here  becomes  viinns.  the  dis- 
tance out,  when  tlie  true  value  of  g  is  found,  will  hQ  A  F  =  A  D  — 
DF-  ^h-^lc-lg. 

For  embankment,  the  process  of  setting  slope  stakes  is  the  same  as 
for  excavation,  except  that  a  rise  in  the  ground  from  the  centre  on 
embankments  corresponds  to  a  fall  on  excavations,  and  vice  vcrsd. 
This  will  be  evident  by  inverting  figurd  45,  which  will  then  represent 


84  LEVELLING. 

an  embankment.     AMiat  was  before  ^  fall  to  Z?,  becomes  now  a  rwe, 
and  what  was  before  a  rise  to  C,  becomes  now  a  fall. 

WHien  tlie  section  is  partly  in-  excavation  and  partly  in  embankment, 
the  method  above  applies  directly  only  to  the  side  which  is  in  excava 
lion  at  the  same  time  that  the  centre  of  the  road-bed  is  in  excavation, 
or  in  embankment  at  the  same  time  that  the  centre  is  in  embank- 
ment. On  the  opposite  side,  however,  it  is  only  necessary  to  make  c 
in  the  expressions  above  minus,  because  its  effect  here  is  to  diminish 
the  distance  out.  The  formula  for  this  distance  out  will,  therefore,  be- 
come ^b  —  2*^  -^  2  y- 


Article  II.  —  Correction  for  the  Earth's  Curvature  and 

FOR  Refraction. 

103.  Let  A  C  (fig.  46)  represent  a  portion  of  the  earth's  surface. 
Then,  if  a  level  be  set  at  A,  tlie  line  of  sight  of  the  level  will  be  the  tap- 
gent  A  D,  while  the  true  level  will  be  A  C.  The  difference  Z)  C  be- 
tween the  line  of  sight  and  the  true  level  is  the  correction  for  the 
earth's  curvature  for  the  distance  ^1  D. 

104.  A  correction  in  the  opposite  direction  arises  from  refraction. 
Refraction  is  the  change  of  direction  which  light  undergoes  in  passing 
from  one  medium  into  another  of  different  density.  As  the  atmos- 
phere increases  in  density  the  nearer  it  lies  to  the  earth's  surface,  light, 
passing  from  a  point  B  to  a.  lower  point  ^4,  enters  continually  air  oJ 
greater  and  greater  density,  and  its  path  is  in  consequence  a  curve 
concave  towards  the  earth.  Near  the  earth's  surface  this  path  may  be 
taKen  as  the  arc  of  a  circle  whose  radius  is  seven  times  the  radius  of 
the  earth.*  Now  a  level  at  A,  having  its  line  of  sight  in  the  direction 
A  D,  tangent  to  the  curve  A  B,  is  in  the  proper  position  to  receive  the 
light  from  an  olyect  at  B  ;  so  that  this  object  appears  to  the  observer 
to  be  at  D.  The  effect  of  refraction,  therefore,  is  to  make  an  object 
appear  higher  than  its  true  position.  Then,  since  the  correction  foj 
the  earth's  curvature  D  C  and  the  correction  for  refraction  D  B  aie  in 
opposite  directions,  the  correction  for  both  will  ha  B  C  =  D  C  —  D  B. 


*  Peirce's  Spherical  Astronomy,  Chap.  X.,  §  125  It  should  be  observed,  how- 
ever, that  the  effect  of  refraction  is  verj'  uncertain,  varjing  with  the  state  of  the 
atmosphere  Sometimes  the  path  of  a  r.i}-  is  even  made  convex  towards  the  earthy 
«nd  sometinies  the  rays  are  refracted  horizontiUy  a^  well  as  yertically. 


I 


earth's  curvature  and  refraction. 


P5 


This  correction  must  be  added  to  the  height  of  any  object  as  deter- 
mined by  the  level. 

105.  Prol>leill.  Given  the  distance  AD  =  D  [Jig.  46),  the  radim 
of  the  earth  A  E  =  R,  and  the  radius  of  the  arc  of  refracted  light  =  7  R, 
'<)  find  the  correction  BC  =  dfor  the  earih's  curvature  and  for  refraction. 


Solution.  To  find  the  correction  for  the  earth's  curvature  D  C,  we 
have,  by  Geometry,  D  C {D  C -{•  2E  C)  =  A  D^  or  D  C  {D  C  +  2  R) 
=  D^.  But  as  Z)  Cis  always  very  small  compared  with  the  diameter 
of  the  earth,  it  may  be  dropped  from  the  parenthesis,  and  we  have 

D  C  X  2  72  =  D-,  or  Z)  C  =  .y-^  .  The  correction  for  refraction  D  B 
may  be  found  by  the  method  just  used  for  finding  D  C,  merely  chang- 
ing  R  into  7  R.  Hence  D  B  =^  ^-j.  .  We  have  then  d  =  B  C  ^ 
DC- DB^  ^ 


J2L 

UR 


or 


d  = 


3D^ 
7R 


By  this  formula  Table  III.  is  calculated,  taking  R  =  20,911,790  ft, 
as  given  by  Bowditch.  The  necessity  for  this  correction  may  be 
avoided,  whenever  it  is  possible  to  set  the  level  midway  between  the 
points  whose  height  is  required.  In  this  case,  as  the  distance  on 
each  side  of  the  level  is  the  same,  the  corrections  will  be  equal,  and 
will  destroy  each  other. 


66 


LEVELLING. 


Article  III.  —  Vertical  Curves. 

106.  Vertical  curves  are  used  to  round  off  the  angles  fonaed  b^ 
the  meeting;  of  two  grades.  Let  A  Cand  CB  (fig.  47)  be  two  grades 
meeting  at  C.  These  grades  are  supposed  to  be  given  by  the  rise  per  sta- 
tion in  uoing  in  some  particuU^r  direction.  Thus,  starting  from  ^1.  the 
grades  of  A  Cand  (^ B  may  be  denoted  respectively  by^  and  9';  that 
is,  (J  denotes  what  is  added  to  the  height  at  every  station  on  A  C,  and 
ij'  denotes  what  is  added  to  the  height  at  every  station  on  CB],hui 
since  CB  is  a  descending  grade,  the  C[uantity  added  is  a  minus  quan- 
tity, and  (/'  will  therefore  be  negative.  The  parabola  furnishes  a  very 
simple  method  of  putting  in  a  vertical  curve. 

107.  Problem.      Given  the  grade  g  of  A  C  [fig.  47),  the  grade  a 
of  C  B,  and  the  number  of  stations  n  on  each  side  of  C  to  the  tangent  points 
A  and  B,  to  unite  these  points  by  a  parabolic  vertical  curve. 


Fig.  47 


Solution.  Let  A  E  B  he  the  required  parabola.  Through  B  and  C 
draw  the  vertical  lines  FK  and  C  H,  and  produce  A  C  to  meet  FK 
in  F.  Through  .1  draw  the  horizontal  line  A  K,  and  join  A  B,  cut- 
ting C  H  in  D.  Then,  since  the  distance  from  C  to  A  and  B  is  meas- 
ured horizontally,  we  have  A  H  =^  H K.  and  consequently  AD  = 
D  B.  The  vertical  line  CD  is,  therefore,  a  diameter  of  the  parabola 
(§  84,  L),  and  the  distances  of  the  curve  in  a  vertical  direction  from 
the  stations  on  the  tangent  A  i^are  to  each  other  as  the  squares  of  the 
number  of  stations  from  A  (^  84,  II.).  Thus,  if  a  represent  this  dis- 
tance at  the  first  station  from  A,  the  distance  at  the  second  station 
would  be  4  a,  at  the  third  station  9  a,  and  at  B^  which  is  2  n  stations 

FB 
from  xV,  it  would  be  4ii^a;  that  is,  FB  =  4n^a,  or  a  =  ^^  .     To  find 

a,  it  will  then  be  necessary  to  find  FB  first.     Through  Cdraw  the 
horizontal  line  C  G   and  we  have,  from  the  equal  triangles  C  F  G  and 


VERTICAL    CURVES.  ^' 

ACH,  FG  =  C II  But  C II  is  the  rise  of  the  first  grade  g  in  the  n 
stations  from  A  to  C;  that  is,  0  ^  =-  n  <j,  or  F  G  =  n  ,j.  G  B  is  also 
the  rise  of  the  second  grade  g'  in  n  stations,  but  since  r/'  is  negative 
(§  106),weinustpat  G'/->^  =  -?ii/'-  Tlicrefore,  FZ^  =  F  G  ■{-  GB 
=  ng  -  ng'.     Substituting  this  value  of  FB  in  the  equation  for  a 


ns  —  n 


:ri 


ive  have  a  =  — -^:      ,  or 

9—9' 


a  = 


4  n 


Tlie  value  of  n  being  thus  determined,  all  the  distances  of  the  curve 
from  the  tangent  .1^;  viz.  a,  4  a,  9  «,  16  a,  &e,  are  known.  Now 
if  ran«i  '/''  be  the  first  and  second  stations  on  the  tangent,  and  verti- 
cal lines  IP  and  2'' P' be  drawn  to  the  horizontal  line  J /if,  the 
height  TP  of  tlie  first  station  above  A  will  be//,  the  height  7''P'  of 
the^sccond  station  above  ^  will  be  2g,  and  in  like  manner  for  suc- 
ceeding stations  we  should  find. the  heights  3</,  4(/,  &c  As  we  have 
already  found  TM  =  a,  T'  M'  =  4  a,  &c.,  we  shall  have  for  the 
heights  of  the  carve  above  the  level  of  A,  MP  =   T P  —  I'M  = 

g (^    ]ji  pi  ^  ']'!  pi  _  T'  M'  =  2g  —  4  a,  and  in  like  manner  for 

the  succeeding  heights  3 ^r  —  9  a,  4^  —  16a,  &c.  Then  to  find  the 
grades  for  the  curve  at  tlie  successive  stations  from  A,  that  is,  the  rise 
of  each  height  over  the  preceding  height,  we  must  subtract  each 
height  from  the  next  following  height,  thus:  {g  —  a)  —0  =  g  —  a, 
{2g-4a)-{g-a)  =  g  -  3  a,  {3  g  -  9  a)  -(2^ -4  a)  =g-5a, 
(4  <7  —  1 6  a)  —  (3  ^  —  9  a)  =  g  —  7  a,  &c.  The  successive  grades  for 
the  vertical  curve  are,  therefore, 

^  S'  —  «5    g  —  3a,    g  —  5a,    g  —  7  a,  &c. 

In  finding  these  grades,  strict  regard  must  be  paid  to  the  algebraic 
signs.  The  results  are  then  general ;  though  the  figure  represents 
but  one  of  the  six  cases  that  may  arise  from  various  combinations  of 
ascending  and  descending  grades.  If  proper  figures  were  drawn  to 
represent  the'  remaining  cases,  the  above  solution,  with  due  attention 
to  the  signs,  would  apply  to  them  all,  and  lead  to  precisely  the  same 
formuloe. 

108.  Examples.     Let  the  number  of  stations  on  each  side  of  Che  3, 
and  let  A  C  ascend  .9  per  station,  and  CB  descend  .6  per  station.   Here 

S-s'       .9- (-.6)       1.5 
n  ^  3,  g  =  .9,  and  g'  =  —.6.      Then,  a  =   -^-  =-      4x3     ~  12 

v_  .12.'>,  and  the  grades  from  A  to  B  will  be 


88  LEVELLING. 

g  —  a  =  .9  —  .125  =  775, 
g  —  3  a  =  .9  —  .375  =  .525, 
g  —  5  a  =  .9  —  .625  =  .275, 
g  —  7  a  =  .9  —  .875  =  .025, 
g  —  9  a  =  .9  —  1.125  =  —  •.225, 
^  ~  11  a  =  .9  —  1.375  =  —  .475. 

As  a  second  example,  let  the  first  of  two  grades  descend  .8  per  s'a 
tion,  and  the  second  ascend  .4  per  station,  and  assume  two  stations  on 
each  side  of  C  as  the  extent  of  the  curve.     Here  g  =  — .8,  g'  =  A, 

and  n  =  2.  Then  a  =  ^'2  —  — s"  ~  —  '^^'  ^"^  ^^^®  ^^^^  grades 
required  will  be 

g—a  =  —  .8—  (— .15)  =  —  .8  +  .15  =  —  .65, 
^  —  3  a  =  —  .8  —  (—  .45)  =  —  .8  +  .45  =  —  .35, 
g  —  5a  =  —  .8  —  ( —  .75)  ==  —  .8  +  .75  =  —  .05. 
^  —  7a  =  —  .8  —  (—  1.05)  =  —  .8  +  1.05  =  +  .25. 

It  will  be  seen,  that,  after  finding  the  first  grade,  the  remaining  grades 
may  be  found  by  the  continual  subtraction  of  2  a.  Thus,  in  the  first 
example,  each  grade  after  the  first  is  .25  less  than  the  preceding  grade, 
and  in  the  second  example,  a  being  here  negative,  each  grade  after 
the  first  is  .3  greater  than  the  preceding  grade. 

109.  The  grades  calculated  for  the  whole  stations,  as  in  the  fore- 
going examples,  are  sufficient  for  all  purposes  except  for  laying  the 
track.  The  grade  stakes  being  then  usually  only  20  feet  apart,  it  will 
be  necessary  to  ascertain  the  proper  grades  on  a  vertical  curve  for 
these  sub-stations.  To  do  this,  nothing  more  is  necessary  than  to  let  g 
and  g'  represent  the  given  grades  for  a  sub-station  of  20  feet,  and  n  the 
number  of  sub-station.s  on  each  side  of  tlie  intersection,  and  to  apply  the 
preceding  formulae.  In  the  last  example,  for  instance,  the  first  grade 
descends  .8  per  station,  or  .16  every  20  feet,  the  second  grade  ascends 
.4  per  station,  or  .08  every  20  feet,  and  the  number  of  sub-stations  io 
200  feet  is  10.      We  have  then  ^  =  —  .16,  g'  =  .08,  and  n  =  10 

—  -16  —  .08  —  .24  -  rr.1  ^       .  1        •         ^T, 

Hence  a  =      ^      -^q      =  —^  =  —  .OOt.     The  first  grade  is,  there 

fore,  g  —  a  =  —  .16  +  .006  =  —  .154,  and  as  each  subsequent  grade 
increases  .012  (§  108),  the  whole  may  be  written  down  without  farther 
trouble,  thus:  —.154,  —.142,  — .130,  —  .118,  —  .106,  —.094,  —.082, 
—  .070,  —.058,  —.046,  —.034,  —.022,  —.010,  +  002,  -f  .014,  +.O^fi. 
+  .038   -h  .050,  +  062,  +  .074. 


ELLVATION    OF    THE    OUTEPt    RAIL    ON    CUKV£ES. 


91 


^      "'^  ft., 
Articlk  IV.  — Elevation  of  the  Outer  Eail  on  Curves.  ^ 

110.  Problem.  Giveti  the  radius  of  a  curve  R,  the  gauge  of  the 
track  g,  and  the  velocity  of  a  car  per  second  v,  to  determine  the  proper  ele- 
vation e  of  the  outer  rail  of  the  curve. 

Solution.  A  car  moving  on  a  curve  of  radius  /?,  with  a  velocity  per  sec- 

ond  =  r,  lias,  by  Mechanics,  a  centrifugal  force  -=  j.  ■  To  counteract 
this  force,  the  outer  rail  on  a  curve  is  raised  above  the  level  of  the 
inner  rail,  so  that  the  car  may  rest  on  an  inclined  plane.  This  eleva- 
tion must  be  such,  that  the  action  of  gravity  in  forcing  the  car  down 
the  inclined  plane  shall  be  just  equal  to  the  centrifugal  force,  which 
impels  it  in  the  opposite  direction.  Now  the  action  of  gravity  on  a 
body  resting  on  an  inclined  plane  is  equal  to  32.2  multiplied  by  the 
ratio  of  the  height  to  the  length  of  the  plane.  But  the  height  of  the 
plane  is  the  elevation  e,  and  its  length  the  gauge  of  the  track  g.  This 
action  of  gravity,  which  is  to  counteract  the  centrifugal  force,  is,  there- 
fore, =  ^^  .    Putting  this  equal  to  the  centrifugal  force,  we  have 


322e 
ifi.2 «        1 


g     ~   1^ 


Hence 


qv* 
e  =    ^ 


32.2  R 

50 
If  we  substitute  for  R  its  value  (§   10)  R  =  ^j^;^  ,  we  have  e  = 

ir  "oA  ?  =  .000G2112  7^2  sin.  D.  If  the  velocity  is  given  in  miles 
^^  ^  ^-^  '  Jfx5280 

per  hour,  represent  this  velocity  by  M,  and  -vve  have  v  =    gQ  ^  qq    ■ 

Substituting  this  value  of  y,  we  find  e  =  .0013361  g  M^  sin.  D.  When 
g  =  4  7,  this  becomes  e  =  .00627966  M^  sin.  D.  By  this  formula 
Table  IV.  is  calculated.  In  determining  the  proper  elevation  in  any 
given  case,  the  usual  practice  is  to  adopt  the  highest  customary  speed 
of  nassenger  trains  as  the  value  of  M. 

111.  Still  the  outer  rail  of  a  curve,  though  elevated  according  to  the 
preceding  formula,  is  generally  found  to  be  much  more  worn  than  the 
inner  rail  On  this  account  some  are  led  to  distrust  the  formula,  and 
to  give  an  increased  elevation  to  the  rail.  So  far,  however,  as  the 
centrifugal  force  is  concerned,  the  formula  is  undoubtedly  correct,  and 
the  evil  in  question  must  arise  from  other  causes,  —  causes  which  are 
not  counteracted  by  an  additional  elevation  of  the  outer  rail.  The 
principal  ofthe.se  causes  is  probably  improper  "  coning"  of  the  wheels. 
Two  wheels,  immovable  on  an  axle,  and  of  the  same  radius,  must,  iC 


90  LE  /ELLING. 

no  slip  is  allowed,  pass  over  equal  spaces  in  a  given  number  of  revo- 
lutions. Now  as  the  outer  rail  of  a  curve  is  longer  than  the  inner  rail, 
the  outer  wheel  of  sucli  a  pair  must  on  a  curve  fall  behind  the  inner 
wheel.  The  first  effect  of  this  is  to  bring  the  flange  of  the  outer  wheel 
against  the  rail,  and  to  keep  it  there.  The  second  is  a  strain  on  the 
axle  consequent  upon  a  slip  of  the  wheels  equal  in  amount  to  the  dif 
ference  in  length  of  the  two  rails  of  the  curve.  To  remedy  this,  con- 
ing of  the  wheels  was  introduced,  by  means  of  which  the  radius  of  the 
outer  wheel  is  in  effect  increased,  the  nearer  its  flange  approaches  the 
rail,  and  this  wheel  is  thus  enabled  to  traverse  a  greater  distance  than 
the  iTiner  w^heel. 

To  find  the  amount  of  coning  for  a  play  of  the  wheels  of  one  inch, 
let  r  and  r'  represent  the  proper  radii  of  the  inner  and  outer  wheels 
respectively,  when  the  flange  of  the  outer  wheel  touches  the  rail.  Then 
r'  —  r  will  be  the  coning  for  one  inch  in  breadth  of  the  tire.  To  ena- 
ble the  wheels  to  keep  pace  with  each  other  in  traversing  a  curve,  their 
radii  must  be  proportional  to  the  lengths  of  the  two  rails  of  the  curve, 
or,  which  is  the  same  thing,  proportional  to  the  radii  of  these  rails.  If 
7t  be  taken  as  the  radius  of  the  inner  rail,  the  radius  of  the  outer  rail 
will  be  72  +  ^,  and  we  shall  have  r  :  r'  ^  R  '.  R  -\-  g.  Therefore,  r  R 
-\-  r  g  ^^  r'  R,  or 

r  —  r  =  _£,  . 
R 

As  an  example,  let  R  =  600,  r  =  1.4,  and  g  =  4.7.     Then  we  have 

1.4  X  4-7 
r'  —  r  —       gQQ  "    —   Oil   ft.      For  a   tire  3.5  in.   wide,  the  coning 

would  be  3. .5  X  .011  =  .038.5  ft.,  or  nearly  half  an  inch.  Wheels 
coned  to  this  amount  would  accommodate  themselves  to  any  curves 
of  not  less  than  600  feet  radius.  On  a  straight  line  the  flanges  of  the 
two  wheels  would  be  equally  distant  from  the  rails,  making  both 
wheels  of  the  same  diameter.  On  a  curve  of  say  2400  feet  radius,  the 
flange  of  the  outer  Avheel  would  assume  a  position  one  fourth  of  an 
inch  nearer  to  the  rail  than  the  flange  of  the  inner  wheel,  which  would 
increase  the  radius  of  the  outer  wheel  just  one  fourth  of  the  necessary 
increase  on  a  curve  of  600  feet.  Should  the  flange  of  the  outer  wheel 
get  too  near  the  rail,  the  disproportionate  increase  of  the  radius  of  this 
wheel  would  make  it  get  the  start  of  the  inner  wheel,  and  cause  the 
flange  to  recede  from  the  rail  again.     If  the  shortest  radius  were  taken 

1.4X  4.7 
as  900  feet,  r  and  g  remaining  the  same,  we  should  have  ?'  —  r  —  — 900"" 


ELEVATION  OF  THE  OUTER  RAIL  ON  CURVES. 


91 


x=  .0073,  and  for  the  coning  of  the  whole  tire  3.5  X  -0073  -  .0256  ft., 
or  about  three  tenths  of  an  inch.  Wheels  coned  to  this  amount  would 
accommodate  themselves  to  any  curve  of  not  less  than  900  feet  radius. 
If  the  wheels  are  larger,  the  coning  must  be  greater,  or  if  the  gauge  of 
the  track  is  wider,  the  coning  must  be  greater.  If  the  play  of  the 
wheels  is  greater,  the  coning  may  be  diminished.  Hence  it  might  be 
advisable  to  increase  the  play  of  the  wheels  on  short  curves,  by  a  slight 
increase  of  the  gauge  of  the  track. 

Two  distinct  things,  therefore,  claim  attention  in  regard  to  the  mo- 
tion of  cars  on  a  curve.     The  first  is  the  centrifugal  force,  which  is 
generated  in  all  cases,  when  a  body  is  constrained  to  move  in  a  cur- 
vilinear path,  and  which  may  be  effectually  counteracted  for  any  given 
velocity  by  elevating  the  outer  rail.     The  second  is  the  unequal  length 
of  the  two  rails  of  a  curve,  in  consequence  of  which  two  wheels  fixed 
on  an  axle  cannot  traverse  a  curve  properly,  unless  some  provision  is 
made  for  increasing  the  diameter  of  the  outer  wheel.     Coning  of  the 
wheels  seems  to  be  the  only  thing  yet  devised  for  obtaining  this  in- 
crease of  diameter.     At  present,  however,  there  is  little   regularity 
either  in  the  coning  itself,  or  in  the  distance  between  the  flanges  of 
wheels  for  tracks  of  the  same  gauge.     The  tendency  has  been  to  di- 
minish the  coning,*  without  substituting  any  thing  in  its  place.     If  the 
wheels  could  be  made  to  turn  independently  of  each  other,  the  whole 
difficulty  would  vanish  ;  but  if  this  is  thought  to  be  impracticable,  the 
present  method  ought  at  least  to  be  reduced  to  some  system. 

*  Bush  and  Lobdell,  extensive  wheel-makers,  say,  in  a  note  published  in  Apple- 
tons'  Mechanic's  Magazine  for  August,  1852,  that  wheels  made  by  them  fcr  the  New 
York  and  Erie  road  have  a  coning  of  but  one  sixteenth  of  an  inch.  This  coning  on 
%  track  of  six  feet  gauge  with  the  c  .her  data  as  given  above,  would  suit  no  ciirva 
•f  less  than  a  mile  radius. 


^.. 


92 


KARTH-WORK. 


CHAPTER   IV. 
EAKTH-WORK. 

Akticlk  I.  —  Prisjioidal  Formula. 

112.  Earth-work  includes  the  regular  excavation  uirI   tinbank 
ment  on  the  line  of  a  road,  borrow-pits,  or  such  additional  excavations 
as  are  made  necessary  when  the  embankment  exceeds  the  regular  ex 
cavation,  and,  in  general,  any  transfers  of  earth  that  require  calcula- 
tion.    We  begin  with  the  prismoidal  formula,  as  this  formula  is  fre- 
quently used  in  calculating  cubical  contents  both  of  earth  and  masonry. 

A  prismoid  is  a  solid  having  two  parallel  faces,  and  composed  of 
prisms,  wedges,  and  pyramids,  whose  common  altitude  is  the  perpen- 
dicular distance  between  the  parallel  fiices. 

113.  Problem.  Given  the  areas  of  the  parallel  faces  B  and  B  , 
the  middle  area  21,  and  the  altitude  a  of  a  prismoid,  to  find  its  solidity  S. 

Solution.  The  middle  area  of  a  prismoid  is  the  area  of  a  section 
midway  between  the  parallel  faces  and  parallel  to  them,  and  the  alti- 
tude is  the  perpendicular  distance  between  the  parallel  faces.  If  now 
b  represents  the  base  of  any  prism  of  altitude  a,  its  solidity  is  ab.  If  6 
represents  the  base  of  a  regular  wedge  or  half-parallelopipedon  of  alti- 
tude a,  its  solidity  is  kab.  Kb  represents  the  base  of  a  pyramid  of 
altitude  a,  its  solidity  is  ^  a  6.  The  solidity  of  these  three  bodies  ad 
mits  of  a  common  expression,  which  may  be  found  thus.  Let  m  rep- 
resent the  middle  area  of  either  of  these  bodies,  that  is,  the  area  of  a 
section  parallel  to  the  base  and  midway  between  the  base  and  top.  In 
the  prism,  m  =  b,  in  the  regular  wedge,  m  =  ^b,  and  in  the  pyramid, 
m  =  ^b.  INIoreover,  the  upper  base  of  the  prism  =  b,  and  the  upper 
base  of  the  wedge  or  pyramid  =  0.  Then  the  expressions  a  b,  ha  &, 
and  kab  may  be  thus  transformed.     Solidity  of 

prism       =     ab  =-  X  &b  =-ib  -\-b -{:  Ab)  =-{b-\-b-{- 4m), 

6  6  6 

wedge     =ia6  =  -X36  =  f.(0  +  6-f2  6)  =-(04-6+4  m), 

6  6  6 

pyramid  =^ab  =  -X2b=-{0-\-b-^b)     =f(0 +  6-1-4 '»j,. 

6  6  6 


EORROW-PITS. 


93 


Hence,  the  solidity  of  either  of  these  bodies  is  found  by  adding  togeth- 
er the  area  of  the  upper  base,  the  area  of  the  lower  base,  and  four 
times  the  middle  area,  and  multiplying  the  sum  by  one  sixth  of  the 
altitude.  Irregular  wedges,  or  those  not  half-parallelopipedons,  may 
be  measured  by  the  same  rule,  since  they  are  the  sum  or  difference  of 
a  regular  wedge  and  a  pyramid  of  common  altitude,  and  as  the  rule 
applies  to  both  these  bodies,  it  applies  to  their  sum  or  difference. 

Now  a  prismoid,  being  made  up  of  prisms,  wedges,  and  pyramids  of 
common  altitude  with  itself,.will  have  for  its  solidity  the  sura  of  the 
solidities  of  the  combined  solids.  But  the  sum  of  the  areas  of  the 
upper  and  lower  bases  of  the  combined  solids  is  equal  to  5  +  B\  the 
sum  of  the  areas  of  the  parallel  faces  of  the  prismoid ;  and  the  sum  of 
the  middle  areas  of  the  combined  solids  is  equal  to  J/,  the  middle  area 
of  the  prismoid.     Therefore  • 

5  =  ^(S  + 5' +  4  37). 
6 


AUTICLE    II.  —  BORROW-PlTS. 

114.  For  the  measurement  of  small  excavations,  such  as  borrow- 
pits,  &c.,  the  usual  method  of  preparing  the  ground  is  to  divide  the 
surface  into  parallelograms  *  or  triangles,  small  enough  to  be  consid- 
ered planes,  laid  off  from  a  base  line,  that  will  remain  untouched  by 
the  excavation.  A  convenient  bench-mark  is  then  selected,  and  levels 
taken  at  all  the  angles  of  the  subdivisions.  After  the  excavation  is 
made,  the  same  subdivisions  are  laid  off  from  the  base  line  upon  the 
oottom  of  the  excavation,  and  levels  referred  to  the  same  bench-mark 
are  taken  at  all  the  angles. 

This  method  divides  the  excavation  into  a  series  of  vertical  prisms, 
generally  truncated  at  top  and  bottom.  The  vertical  edges  of  these 
prisms  are  known,  since  they  are  the  differences  of  the  levels  at  the 
top  and  bottom  of  the  excavation.  The  horizontal  section  of  the 
prisms  is  also  knoAvn,  because  the  parallelograms  or  triangles,  into 
which  the  surface  is  divided,  are  always  measured  horizontally. 

11.5.  Problem.      Given  the  edges  h,  hi ,  and  ho  ,  to  find  the  solidity 


•  If  the  ground  is  divided  into  rectangles,  as  is  generally  done,  and  one  side  b« 
made  27  feet,  or  some  multiple  of  27  feet,  the  contents  may  be  obtained  at  once  in 
rubic  yards,  by  merely  omitting  the  factor  27  in  the  calculation. 


94 


EARTH-WORK. 


S  of  a  veitical  prism,  whether  truncated  or  not.  whose  horizontal  section  ti 
o  triangle  of  given  area  A. 


Fig.  48 


Solution.  "Wlicn  the  prism  is  not  truncated,  we  have  h  =  h^  =  k^' 
The  ordinary  mle  for  the  solidity  of  a  prism  gives,  therefore,  S  =  Ah 
■^  A  X  b  {h  +  hi  -{-  hr,).  When  the  prism  is  truncated,  let  ABG- 
F  G  H {i\g.  48)  represent  such  a  prism,  truncated  at  the  top.  Through 
the  lowest  point  A  of  the  upper  face  draw  a  horizontal  plane  A  D  E 
cutting  off  a  pyramid,  of  which  the  base  is  the  trapezoid  B  D  E  C,  and 
the  altitude  a  perpendicular  let  fall  from  A  on  D  E.  Represent  this 
perpendicular  by  p,  and  we  have  (Tab,  X.  52)  the  solidity  of  the  pyra- 
mid =  ^px  BDEC  ==\pxDExh{BD^  C  E)  =  ^pX 
DE  X  ^  {BD  -\-  CE)  =  A  X  h  [BD  +  CE),  since  hp  X  DE 
=  A  D  E  =  A.  But  I  {BD  -\-  CE)  is  the  mean  height  of  the  verti- 
cal edges  of  the  truncated  portion,  the  height  at  A  being  0.  Hence 
the  formula  already  found  for  a  prism  not  truncated,  will  apply  to  the 
portion  above  the  plane  ^  Z>  £",  as  well  as  to  that  below.  The  same 
reasoning  would  apply,  if  the  lower  end  also  were  truncated.  Hence, 
for  the  solidity  of  the  Avhole  prism,  whether  truncated  or  not,  we  have 

S=AXhih  +  h,+  h.). 


116.  Problem.  Given  the  edges  h,  h^,  hn,  and  A3,  to  Ji7id  tU 
solidity  S  of  a  vertical  prism,  ivheiher  truncated  or  not,  whose  horizoUat 
section  is  a  parallelogram,  of  given  area  A. 


BORROW-PITS. 


9fi 


Solution.    Let  B  H  (fig.  49)  represent  such  a  prism,  whether  trim 
cated  or  not,  and  let  the  plane  BFHD  diviie  it  into  two  triangular 


Fig.  49 


prisms  AFH  and  C F H.  The  horizontal  section  of  each  of  these 
prisms  will  be  ^  A,  and  if  A,  h^ ,  h^ ,  and  h^  represent  the  edges  to  which 
they  are  attached  in  the  figure,  we  have  for  their  solidity  (§  115) 
A  FH  =^A  X  k  i^i-^  h  +  h).  and  CFH  =  ^A  X  ^  (^i  +  h  + 
^g).  Therefore,  the  whole  prism  will  have  for  its  solidity  S  =  ^  A  X 
^  {h  +  2/tji  +  112  +  2  A3).  Let  the  whole  prism  be  again  divided  b} 
the  plane  AE  G  C  into  two  triangular  prisms  BEG  and  D  E  G 
Then  we  have  for  these  prisms,  B  E  G  =  hA  X  ^  {^^  +  ^h  +  h)^ 
and  D  E  G  =  h  A  X  J  (^  +  ^'2  +  '^3)5  and  for  the  whole  prism,  S  — 
^A  X  ^  (2  A  +  /ij  +  2  /<2  +  h).  Adding  the  two  expressions  found 
for  S,  we  have  2  S  =  ^  A  {h  -^  h^  +  h^  -\-  Jh),  or 

^  S=A  X  i{h-{-h,  +  h.  +  h,). 

It  will  be  seen  by  the  figure,  that  h  {h  +  ho)  =  KL  =  h  {K  +  fh), 
or  h  -\-  kz  =  hi  -{-  h^ .  The  expression  for  S  might,  therefore,  be  re- 
duced to  S  =  A  X  k  i^  +  h),  or  S  =  A  X  ^  {hi  +  h^).  But  as 
the  ground  surfaces  A  B  CD  and  E F  GHare  seldom  perfect  planes, 
it  is  considered  l>etter  to  use  the  mean  of  the  four  heights,  instead  of 
the  mean  of  two  diagonally  opposite. 

117.  Corollary.  When  all  the  prisms  of  an  excaA-ation  have 
ilic  same  horizontal  section  A,  the  calculation  of  any  number  of  them 


06 


KARTH-WOKK 


may  be  performed  by  one  operation.    Let  figure  50  be  a  plan  ot  such 
an  excavation,  the  heights  at  the  angles  being  denoted  by  a,  Oi ,  Oo,  ^ 


a, 


«*« 


\h3 


d* 


bs 


\c 


C/ 


r^ 


C3 


Pd.  Ca 


d 


d> 


ds 


a> 


Fig.  50. 


6i ,  &c.  Then  the  solidity  of  the  whole  will  be  equal  to  \A  multi 
plied  by  the  sum  of  the  heights  of  the  several  prisms  (§  116).  Into 
this  sum  the  corner  heights  a,  Oo ,  h^h^,  Cj,,  </,  and  d^  will  enter  but 
once,  each  being  found  in  but  one  prism  ;  the  heights  01,^4,  c,  di,  do, 
and  rfj  will  enter  ticice,  each  being  common  to  two  prisms  ;  the  heights 
fe. ,  bj,  and  t'4  will  enter  three  times,  each  being  common  to  three 
prisms;  and  the  heights  ioj^ijCo,  and  c^  will  enter  four  times,  each 
being  common  to  four  prisms.  If,  therefore,  the  sum  of  the  first  set  of 
heights  is  represented  by  Si ,  the  sum  of  the  second  by  So ,  of  the  third 
by  S3 ,  and  of  the  fourth  by  s^ ,  we  shall  have  for  the  solidity  of  all  tho 
pnsms 

>S  =  I  J.  (si  +  2  So  +  3  S3  +  4  S4). 


Article  III.  —  Excavation  and  Embankment. 

118.  As  embankments  have  the  same  general  shape  as  excavations, 
it  will  be  necessary  to  consider  excavations  only.  The  "simplest  case 
is  when  the  ground  is  considered  level  on  each  side  of  the  centre  line. 
Figure  51  represents  the  mass  of  earth  between  two  stations  in  an  ex- 
cavation of  this  kind.  The  trapezoid  G  B  F  H  is  a  section  of  the 
mass  at  the  first  station,  and  Gi  Bi  F^  H^  a  section  at  the  second  sta- 
tion; AE  \s  the  centre  height  at  the  first  station,  and  A^  E^  the  centre 
height  at  the  second  station  ;  HffiFiFis  the  road-bed,  G  Gi  B^  B  the 


CENTRE    HEIGHTS    ALOJSE    GU^EN. 


9*: 


surface  of  the  ground,  and  G  Gi  H^  11  and  BB^F^F  the  planes  form- 
ing the  side  slopes.  This  solid  is  a  prismoid,  and  might  be  calculated 
bv^the  prismoidal  formula  (§  113).  The  following  metaod  gives  the 
same  result. 


A.     Centre  Ileifjhts  alone  given. 

119.  Problem.  Given  the  centre  heiyhts  c  and  Cj ,  the  width  of  the 
road-hed  6,  the  slope  of  the  sides  s,  and  the  length  of  the  section  I,  to  find 
*he  soliditij  S  of  the  excavation. 


Fig.  51. 


}sol-iiion.  Let  c  be  the  centre  height  at  A  (lig.  51)  and  Cj  the  height 
af,  X. .  The  slope  s  is  the  ratio  of  the  base  of  the  slope  to  its  perpen- 
dicular height  (§  102).  We  have  then  the  distance  out  ^  B  =  ^6  + 
sc,  and  the  distance  out  A^B^  ^  \h -\-sci{\  102).  Divide  the  whole 
mass  into  two  equal  parts  by  a  vertical  plane  A  Ai  E^  E  drawn 
through  the  centre  line,  and  let  us  find  first  the  solidity  of  the  right- 
hand  half.  Through  B  draw  the  planes  BEE^,  BA^Ei,  and 
B^jFi,  dividing  the  half-section  into  three  quadrangular  pyramids, 
having  for  their  common  vertex  the  point  Z5,  and  for  their  bases  the 
planes  AA^EiE,  E Ey  Fi  F,  and  AiBiF^Ei.  For  the  areas  of  these 
bases  we  have 


Areaof  ^  Ji^i^ 
"  "  EEiFiF 
"    "  A,B,F,E, 


=  iEEi  X  {AE-{-  A^E,)  = 
^EFx  EE,  = 

=^^A,E,X{E,F,+A,Bi):^^{bc,^sc,*); 


^/(c  +  Ci), 
hbl. 


and  lor  tlie  perpendiculars  from  the  vertex  B  on  these  bases,  produced 
when  necesyarv. 


98  EARTH-WORK. 

Perpendicular  on  A  A^E^E     =  A  B   —  1 6  -f  o  c, 
''  EExF^F      =  AE   ^  c, 
"  «  A^B^F.Ei  =  EEi  =  I. 

Then  (Tab.  X.  52)  the  solidities  of  the  three  pyramids  are 

B-AA,E,E    =|(i6  +  sc)  X  ^/(c  +  cO=|/(i6c-f-^6c,-^ 

B-EE^l\F    =\cY.\hl  ^'llbc, 

B-A^B,F,E,=  II  X  h  (^Ci  +  sO  =U(6ci+sci2). 

Their  sum,  or  the  solidity  of  the  half-section,  is 

LS  =  \l[lh{c-\-  Ci)  +  s  (c^  +  Ci2  +  cci)l. 

Therefore  the  solidity  of  the  whole  section  is 

^S-  -  i  /  [i  Mc  +  cx)  -f  s  (c^  +  c,-'  +  cc,)J, 
or 

^  5  =  i  /  [6  (c  +  c)  4-  I  s  (c'  +  Ci^  -f  c  c,) J 

When  the  slope  is  1^  to  1,  s  =  i,  and  the  factor  fs  =   I  may  be 
dropped. 

120.  Problem.  To  Jind  the  solidity  S  of  any  number  n  of  succes- 
sive sections  of  equal  length. 

Solution.  Let  c,  Ci,C2,C3,  &c.  denote  the  centre  heights  at  the  suc- 
cessive stations.     Then  we  have  (§  119) 

Solidity  of  first  section      =  ^l[b  {c   +  Ci)  -f  f  s  (c^    +  Cj^  +  c  c^)], 
"       «  second  section  =  ^ Z  [6  (ci  +  Co)  +  | s  (cj*  +  Co-  +  c^  Co)], 
«       "   third  section     =  |/  [6  (cg  +  Cg)  +  |  s  (ca^  +  Ca^  +  C0C3)], 
&c.  &c. 

For  the  solidity  of  any  number  n  of  sections,  we  should  have  ^l  mul- 
tiplied by  the  sum  of  the  quantities  in  n  parentheses  formed  as  those 
iust  given.  The  last  centre  height,  according  to  the  notation  adopted, 
will  be  represented  by  c,  and  the  next  to  the  last  by  c„_i.  Collect- 
ing the  terms  multiplied  by  b  into  one  line,  the  squares  multiplied  by 
I  s  into  a  second  line,  and  the  remaining  terms  into  a  third  line,  we 
have  for  the  solidity  of  n  sections 

^^     S=hl  6  (c4- 2f:  -f  2r, -f  2c3 +  2c„_i  +  c„) 

4.  |S   (c2+2Ci2  4-2C22  +  2C32....  +2c2„_l  +  C»„) 
+  I  S  (C  Ci  +  Ci  Co  +  C2  C3  +  ^a  C4 +  c„-  1  On). 

When  s  =  I ,  the  factor  f  s  =  1  may  be  dropped. 


CENTRE    AND    SIDE    HEIGHTS    GIVEN. 


99 


Example.  Given  /  =  100,  6  =  28,  s  =  i ,  and  the  stations  and  cen- 
tre heights  as  set  down  in  the  first  and  second  columns  of  the  annexed 
table.  ''The  calculation  is  thus  performed.  Square  the  heights,  and 
set  the  squares  in  the  third  column.  Form  the  successive  products 
c  ci ,  Ci  C2 ,  &c.,  and  place  them  in  the  fourth  column.  Add  up  the  last 
three  columns.  To  the  sum  of  the  second  column  add  the  sum  itself, 
minus  the  first  and  the  last  height,  and  to  the  sum  of  the  third  column 
add  the  sum  itself,  minus  the  first  and  the  last  square.  Then  86  is  the 
multiplier  of  b  in  the  first  line  of  the  formula,  592  is  the  second  line, 
since  §  s  is  here  1,  and  274  is  the  third  line.  The  product  of  86  by  b 
=  28  is  2408,  and  the  sum  of  274,  .592,  and  2408  is  3274.  This  mul- 
tiplied by  |/  --=  50  gives  for  the  solidity  163.700  cubic  feet. 


Station. 

c. 

c-i. 

CCi. 

0 

9 

4 

1 

4 

16 

8 

2 

7 

49 

28 

3 

6 

36 

42 

4 

10 

100 

60 

5 

1 

49 

70 

6 

6 

36 

42 

7 

4 

16 

24 

46 

306 

274 

40 

286 

592 

86 

592 

2408 

28 

2)3274 

2408 


163700. 


B.    Centre  and  Side  Heights  given. 

121.  When  greater  accuracy  is  required  than  can  be  attained  by  Ae 
preceding  method,  the  side  heights  and  the  distances  out  (§  102)  are 
introduced.  Let  figure  52  represent  the  riglit-hand  side  of  an  excava 
tion  between  two  stations.  AAi  By  B  is  the  ground  surface  ;  AE  =^  c 
and  A^Ei  =  Ci  are  the  centre  heiglits  ;  B  G  =  h  and  C,  Gi  =  hi ,  the 
side  heights  ;  and  d  and  d^ ,  the  distances  out,  or  the  horizontal  distan- 
ces of  B  and  Bi  from  the  centre  line.  The  whole  ground  surface 
may  sometimes  be  taken  as  a  plane,  and  sometimes  the  part  on  each 
side  of  the  centre  line  may  be  so  taken  ;  *  but  neither  of  these  suppo- 


*  It  is  easy  in  any  given  case  to  ascertain  whether  a  surface  like  A  Ai  Bi  £  is  a 


iOO  EARTH-WORK. 

sitions  is  sufficiently  accurate  to  serve  as  the  basis  of  a  general  mciiiod. 
In  most  cases,  however,  we  may  consider  the  surface  on  each  side  of 
the  centre  line  to  be  divided  into  two  triangular  planes  by  a  diagonal 
passing  from  one  of  the  centre  heights  to  one  of  the  side  heights.  A 
ridge  or  depression  will,  in  general,  determine  which  diagonal  ought 
to  be  taken  as  the  dividing  line,  and  this  diagonal  must  be  noted  in 
the  field.  Thus,  in  the  figure  a  ridge  is  supposed  to  run  from  B  to 
^4.1,  from  which  the  ground  slopes  downward  on  each  side  to  A  and 
Bi .  Instead  of  this,  a  depression  might  run  from  A  to  B^ ,  and  the 
ground  rise  each  way  to  A^  and  B.  If  the  ridge  or  depression  is  very 
marked,  and  does  not  cross  the  centre  or  side  lines  at  the  regular  sta- 
tions, intermediate  stations  must  be  introduced  to  make  the  triangular 
planes  conform  better  to  the  nature  of  the  ground.  If  the  surface 
happens  to  be  a  plane,  or  nearly  so,  the  diagonal  may  be  taken  in 
either  direction.  It  will  be  seen,  therefore,  that  the  following  method 
is  applicable  to  all  ordinary  ground.  When,  however,  the  ground  is 
very  irregular,  the  method  of  §  127  is  to  be  used. 

122.  Problem.  Given  the  centre  heights  c  and  c^ ,  the  side  heights 
on  the  right  h  and  h^ ,  on  the  left  h'  and  h\  ,  the  distances  out  on  the  right 
d  and  d^ ,  on  the  left  d'  and  d'l ,  the  icidth  of  the  road-bed  b,  the  length  of  the 
section  /,  and  the  direction  of  the  diagonals,  to  find  the  solidity  S  of  the 

excavation. 

Solution.  Let  figure  52  represent  the  right-hand  side  of  the  excava- 
tion, and  let  us  suppose  first,  that  the  diagonal  runs,  as  shov,n  in  the 
figure,  from  B  to  Ai-  Through  B  draw  the  planes  B  E  E^,  B  A^Ei, 
and  BEiFi,  dividing  the  half-section  into  three  quadrangular  pyra- 
mids, having  for  their  common  vertex  the  point  B,  and  for  their  bases 
the  planes  A  A^  E^E,  E  E^  F^  F,  and  A,  B,  F^  E, .  For  the  areas 
of  these  bases  we  have 

Areaof^^i^iJ^;   =  ^  E  E,  x{AL-]-A^E,)     -|/(c-fc,), 
"    ^'EE.FiF    =EFxEEi  =^l^h 

"    »  A,  B,  F,  E,  =  ^  A,  E^xdi  +  k  ^i  F^Xh,  =  ^d,c,  -\-  ibh, , 

and  for  the  perpendiculars  from  the  vertex  B  on  these  bases,  produced 
when  necessary, 

plane  ;  for  if  it  is  a  plane,  the  descent  from  A  to  B  will  be  to  the  descent  from  Ai  to 
Bi ,  as  the  distance  out  at  the  first  station  is  to  the  distance  out  at  the  second  sta- 
tion, that  is,  c  —  h:ci  —  hi  =  d:di.  K  we  had  c  =  9,  A  =  6,  fi  =  12,  «!  =  8, 
d  =  24,  and  di  =  27,  the  formula  would  give  3  :  4  =  24  :  27  which  shows  that  tho 
lurface  is  not  a  plane. 


CENTRE    AND    SIDE    HEIGHTS    GIVEN 

Perpendicular  on  A  A^^  E^  E  —  E  G  =  d, 
"  E  E,  F,  F  =^  BG  ^K 
"  A,B,F,E,  =  EE,  -/. 


10) 


A  I 


Fig.  52. 


Then  (Tab.  X.  52)  the  solidities  of  the  three  pyramids  arc 

B-AA,E,E     =  :^d  X  ^-Mc  +  ci)  =  |/ (c?c  +  c/c,), 

B-EE,F,F     =  I  A  X  ^  '^^  =llbh, 

B-A,B,F,E,  =kl  X  h{dic,  +  ^bh,)  =  U(^iCi+^6A,). 

Their  mm,  or  tlic  solidity  of  the  half-section,  is 

ll{dc-\-  d,c^  +  dc,  +  hh  +  hhK).  (1) 

Next,  suppose  that  the  diagonal  runs  from  A  to  B^ .  In  this  case, 
through  B,  draw  the  planes  B,  E,  E,  B,  A  E,  and  B^EF  {not  rep- 
resented  in  the  figure),  dividing  the  half-section  again  into  three 
quadrangular  pyramids,  having  for  their  common  vertex  the  point 
Bi ,  and  for  their  bases  the  planes  A  A,  E^  E,  E E^  F^  F,  and  A  B FE 
For  the  areas  of  these  bases  -vve  have 

Area  of  ^  ^1 ,  ^,  ^  =  U^^ ^:  X  {A  E -{-  A^E^)  ^  ^l  {^  +  c^), 
"     ''  EE^FiF   =EFx  EE^  =h^h 

»    ''  ABFE      =^AExd-{-^EFxh    =^dc-\-  ^bh; 

and  for  the  perpendiculars  from  Bi  on  these  bases,  produced  when 
necessary, 


102  EARTH- WORK. 


Perpendicular  on  A  Ai  E^  E  =  E^  G^  =  d^ 
«  «  ABFE     =  E  El    =  I. 


Tiin  {Tab.  X.  52)  the  solidities  of  the  three  pyramids  are 

Bi-AAiEiE=  ^di  X  hl{c  +  ci)     =hl{chc-\-diCi). 

Bi-EEiF^F  =  ^hi  X  k^'l  =  lib  hi, 

Bi-  ABFE     =11  X  ^{dc  +  ^bh)  =  \l{dc-\-  \bh). 

Their  sum,  or  the  solidity  of  the'  half-section,  is 

\l{dc  +  diCi  +  dic  -\-bhi  +  hbh).  (2) 

We  have  thus  found  the  solidity  of  the  half-section  for  both  direc 
tions  of  the  diagonal.     Let  us  now  compare  the  results  (1)  and  (2), 
and  express  them,  if  possible,  by  one  formula.    For  this  purpose  let 
(1)  be  put  under  the  form 

ll[dc  +  diCi-^dci  J^lb[h+hi  4-^)1, 

and  (2)  under  the  form 

il[dc  +  d,ci  +  dic-^\b  [h  +  hi  +  hi)\. 

The  only  difference  in  these  two  expressions  is,  that  dci  and  the  last 
h  in  the  first,  become  di  c  and  Aj  in  the  second.  But  in  the  first  case, 
c,  and  h  are  the  heights  at  the  extremities  of  the  diagonal,  and  d  is  the 
distance  out  corresponding  to  h ;  and  in  the  second  case,  c  and  hi  are 
the  heights  at  the  extremities  of  the  diagonal,  and  di  is  the  distance 
out  corresponding  to  hi.  Denote  the  centre  height  touched  bij  the  diagonal 
by  C,  the  side  height  touched  by  the  diagonal  by  H,  and  the  distance  out  cor- 
responding to  the  side  height  H  by  D.  We  may  then  express  both  c/c, 
and  dichy  D  C,  and  both  h  and  hi  by  //;  so  that  the  solidity  of  the 
half-section  on  the  right  of  the  centre  line,  whichever  way  the  diago- 
nal runs,  may  be  expressed  by 

\l[dc^diCi  -^DC-\-^b[h-^hi  +  H)\.  (3) 

To  obtain  the  contents  of  the  portion  on  the  left  of  the  centre  line, 
we  designate  the  quantities  on  the  left  by  the  same  letters  used  for  cor- 
responding quantities  on  the  right,  merely  attaching  a  (')  to  them  to 
distinguish  them.  Thus  the  side  heights  are  h'  and  h'l ,  and  the  dis- 
tances out  d'  and  d'l ,  while  Z),  C,  and  H  become  Z)',  C,  and  H'. 
The  solidity  of  the  half-section  on  the  left  may  therefore  be  taken  di- 
rectly from  (3),  which  will  become 


CENTRE    AND    SIDE    HEIGHTS    GIVEN. 


io;j 


Finally,  by  uniting  (3)  and  (4),  ^vc  obtain  ilie  following  formula  for 
the  solidity  of  the  whole  section  between  two  stations 
j^      ^-^  U{{d-\-d')c-^r{d,^cl\)c,^DC-\-D'C<-\-'^h{h-{- 

Example.  Given  /  =  100,  6  =  18,  and  the  remaining  data,  as  ar 
langcMl  in  the  first  six  columns  of  the  following  tabic.  The  first  col- 
i.nn  gives  the  stations  ;  the  fourth  gives  the  centre  heights,  namely, 
c  --  13.6  luul  ci  =-  8  ;  the  two  columns  on  the  left  of  the  centre  heights 
give  the  side  heights  and  distances  out  on  the  left  of  the  centre  line  of 
tlie  road,  and  the  two  columns  on  the  right  of  the  centre  heights  give 
the  side  heights  and  distances  out  on  the  right.  The  direction  of  the 
diagonals  is  marked  by  the  oblique  lines  drawn_^from  h'  =  8  to  Cj  =-  8 
and  from  c  =^   13. 0  lo  //^  ^=  12. 


Sta. 

0 

I 

d'. 

21 
15 

8\ 
4 

c. 

h. 

10 
^^12 

d. 

24 

27 

'  d  +  d'. 

(d  +  d^)c. 

D'  C 
1G8 

DC. 

13.6  \ 

^  8.0 

45 

42 

612 
336 

367.2 

12 

12            168 
20            367.2 
54  X  9  =       486 

• 

6)1969.2( 

3 

32820. 

To  apply  the  formula,  the  distances  out  at  each  station  are  added 
together,  and  their  sum  placed  in  the  seventh  column ;  these  sums, 
multiplied  by  the  respective  centre  heights,  are  placed  in  the  eightli 
column  ;  the  product  off/'  ==  21  (which  is  the  distance  out  correspond- 
inc^  to  the  side  height  touched  by  the  left-hand  diagonal)  by  c,  =  8 
(which  is  the  centre  height  touched  by  the  same  diagonal)  is  placed 
m  the  ninth  column,  and  the  similar  product  of  c/j  =  27  by  c  =  13.6 
is  placed  in  the  last  column.  The  terms  in  the  formula  multiplied  by 
^  b  are  all  the  side  heights,  and  in  addition  all  the  side  heights  touched 
by  diagonals,  or  8  +  4  +  10  +  12  +  8  +  12  =  54.  Then  by  sub- 
stitution  in  the  formiila,  we  have  S  ==  h  X  100  (612  +  336  +  168  + 
867.2  +  9  X  54)  =-  32,820  cubic  feet.* 


*  The  example  here  given  is  the  same  as  that  calculated  in  Mr.  Borden's  "  Sya- 


104  EARTH -WORK. 

By  applying  the  rule  given  in  the  note  to  §  !'21,  we  see  that  the  sar- 
face  on  the  left  of  the  centre  line  in  the  preceding  example  is  a  plane  • 
since  13.6  —  8  :  8  —  4  =  21  :  15.  The  diagonal  on  that  side  might, 
therefore,  be  taken  either  way,  and  the  same  solidity  would  be  ob- 
tained. This  may  be  easily  seen  by  reversing  the  diagonal  in  this  ex- 
ample, and  calculating  the  solidity  anew.  The  only  parts  of  the  for- 
mula affected  by  the  change  are  D'  C  and  ^b  H'.  In  the  one  case 
the  sum  of  these  terms  is  21  X  8  +  9  X  8,  and  in  the  other  15  X  13.6 
+  9X4,  both  of  which  arc  equal  to  240. 

123  Problem.  To  find  the  solidity  S  of  any  number  n  of  succes- 
sive sections  of  equal  length. 

Solution.  Let  c,  Cj  ,  Co ,  c^,  &c.  be  the  centre  heights  at  the  succes- 
sive stations;  /(.  lii ,  h., ,  h^  ,  &c.  the  right-hand  side  heights;  h',  li\  ,  A'o  , 
Zi'o ,  .fcc.  the  left-hand  side  heights ;  (/,  t/j ,  c/., ,  d^  ,  &c.  the  distances  out 
on  the  right ;  and  t/',  d\  ,  d'^ ,  d'^ ,  &c.  the  distances  out  on  the  left. 
Then  the  formula  for  the  solidity  of  one  section  (§  122)  gives  for  thp 
solidities  of  the  successive  sections 

\l[{d-\-d')cJr{<-h  +^'i)c,  ^DC+D'  C'-\-hb{h  +  h,-^  H-\. 
h[-\-h\^H% 

\l[[d^J^d\)c,  ^{d.-\-d>.)c.  +  D,  Ci  +  D\C\-^^b{h^  +A2-H 
ZTi  +  A'.  +  A'o  +  H'OJ, 

G I  \{dn  +  J'o)  c,  +  ((/3  +  c/'a)  C3  +  D.  a  +  Z)'.  C'2  +  i  6  {h.  +  A3  -f 
H.-i-h',  +  h>,.^H'.)l 

"^nd  so  on,  for  any  number  of  sections.  For  the  solidity  of  any  num- 
ber n  of  sections,  we  should  have  g  /  multiplied  by  the  sum  of  n  paren- 
theses formed  as  those  just  given.     Hence 

^  a5-  I  /  (c?+  cZ')  c  +  2  {d,-\- d\)cy-\-2  {d.,  +  f/'^)  Co  . . . -f  {d„  +  d'„)  c„ 
+  DC+  D'C>  +  Z)iCi  -I-  D\  C\  +  B.C.  +  D'.C.  +  &c. 

4-  ^  6  i  /i  +  2  Ai  +  2  /?., +  Ih,  +  ^+  i/i  +  ZTo  +  &c. 

I  +  /i'+  2 /t'i+  2  A'o . . .  +  It'n  +  H'-\-H\-\-H'.  +  &.C. 


tem  of  Useful  Fonnulne,  &c  ,"  page  187.    It  will  be  seen,  that  his  calculation  make? 
the  solidity  32,460  cubic  feet,  which  is  360  cubic  feet  less  than  the  result  above. 
This  difference  is  owing  to  the  omission,  by  Mr.  Borden's  method,  of  a  pyramid  in- 
closed by  the  four  pyramids,  into  which  the  upper  portion  of  the  right-hand  hall 
section  is  by  that  method  divided. 


CKNTRE    AND    SIDE    HEIGHTS    GIVEN. 


105 


Example.     Given  /  =  100,  b  =  28,  and  the  remaining  data  as  given 
in  the  first  six  columns  of  the  following  table. 


'Sta. 
0 

d'. 

k'. 

c. 

h. 

d. 

17 

2            2 

2 

17 

1 

18.5 

3       >4_ 

5 

21.5 

2 

20 

4-^  ^5^ 

^6 

23 

3 

23 

6 -^^6  ..^ 

'"•s 

26 

4 

21. .5 

5-^0,^0 

>7 

24.5 

5 

20 

4  -^U^  G  / 

A 

20      , 

6 

15.5 

1-^ 

i^ 

3 

18.5 

d  +  d' 


25 

22 

90 


69 
102 


171  X  14 


35 

30 

37 

T02 

2394 


2394 
6)6212 


103533  cubic  feet. 


The  data  in  this  table  are  arranged  precisely  as  in  the  example  for  cal- 
culating one  section  (§  122),  and  the  remaining  columns  are  calculated 
as  there  shown.  Then,  to  obtain  the  first  line  of  the  formula,  add  all 
the  cumbers  in  the  column  headed  {d-\-  d')  c,  making  1389,  and  after- 
wards all  the  numbers  except  the  first  and  the  last,  making  1185. 
The  next  line  of  the  formula  is  the  sum  of  the  columns  D'  C  and 
D  C,  which  give  respectively  605  and  639.  To  obtain  the  first  line  of 
the  quantities  multiplied  by  \b^  add  all  the  numbers  in  column  A, 
making  35,  next  all  the  numbers  except  the  first  and  the  last,  making 
30,  and  lastly  all  the  numbers  touched  by  diagonals  (doubling  any  one 
touched  by  two  diagonals),  making  37.  The  second  line  of  the  quan- 
tities multiplied  by  ^6  is  obtained  in  the  same  way  from  the  column 
marked  A'.  The  sum  of  these  numbers  is  171,  and  this  multiplied  by 
16=14  gives  2394.  "We  have  now  for  the  first  line  of  the  formula 
1389  +  1185,  for  the  second  605  +  639,  and  for  the  remainder  2394. 

100 
By  adding  these  together,  and  multiplying  the  sum  by  5/  =  -g-  ,  we 

get  the  contents  of  the  six  sections  in  feet. 

124.  When  the  section  is  partly  in  excavation  and  partly  in  embank- 
ment, the  preceding  formula?  are  still  applicable  ;  but  as  this  applica- 
tion introduces  minus  quantities  into  the  calculation,  the  following 
method,  similar  in  principle,  is  preferable. 

125.  Problem*      Given  the  ividlhs  of  an  excavation  at  the  road-bed 

6 


106 


EARTH-WORK. 


AF  =  w  and  Ai  F,  =  Wi  {Jig.  53),  the  side  heights  h  and  h^.the  lenfftk 
of  the  section  /,  arid  the  direction  of  the  diagonal,  to  find  the  solidity  S  of 
the  excavation,  when  the  section  is  partly  in  excavation  and  partly  in  em- 
bankment. 


Fig.  53 


Solution.     Suppose,  first,  that  the  surface  is  divided  into  two  trian 
gles  by  the  diagonal  B  A^.      Through  B  draw  the  plane  BA^F,, 
dividing  that  part  of  the  section  which  is  in  excavation  into  two  pyra- 
mids B-AAiFiF  and  B-AiB^  F^ ,  the  solidities  of  which  are 

B  -  A  Ai  F,  F  =  I  h  X  k  ^  {lo  +  ivi)  =  ll{ioh  -\-  wi  h), 

B-AiBiFi     =^lx^ioihi  =llwihi. 

The  whole  solidity  is,  therefore, 

S  =  kl  {wh  -\-  ivi  Aj,  +  it'i  h). 

Next,  suppose  the  dividing  diagonal  to  run  from  Ato  Bi.  Through 
Bi  draw  a  plane  BiAF  (not  represented  in  the  figure),  dividing  the 
excavation  again  into  two  pyramids,  of  which  the  solidities  are 

Bi-AAiF^F^^hi  X  hl{io-\-Wi)  =  \l{ioh  +  ^o^h)y 
Bi-ABF        =^lxh^h  =11  wh. 

The  whole  solidity  is,  therefore, 

S  =  ll{wh  +  Wihi  +  lohi). 

The  only  diff'erence  in  these  two  expressions  is.  that  iVj  h  in  the  first 
becomes  v;/«i  in  the  second.  But  in  the  first  case  the  diagonal  touch- 
es io\  and  h,  and  in  the  second  case  it  touches  iv  and  h^.  If,  then,  we 
designate  the  width  touched  by  the  diagonal  by  W,  and  the  height 
touched  by  the  diagonal  by  H,  we  may  express  both  Wi  h  and  tv  h^  by 
WH;  so  that  the  solidity  in  either  case  may-be  expressed  by 


CENTRE    AND    SIDE    HEIGHTS    GIVEN. 


lOT 


S^ll{ivh  +  iv,h,  +  WII). 


Corollary.  When  several  sections  of  equal  length  succeed  one 
another,  the  whole  may  be  calculated  together.  For  this  purpose,  the 
preceding  formula  gives  for  the  solidities  of  the  successive  sections 

ll{ivh     +  it'iAj  +  IF//), 

ll(w,h,   +  H',/2o+    TF1//1), 

and  so  on  for  any  number  of  sections.     Hence  for  the  solidity  of  any 
number  n  of  sections  we  should  have 

E^  S=ll{ivJi  +  2ii\  /ii  -f-  2  1^3  /to  ....  4-  Wn  hn  -f  WH -\-  Wi  H^  -I- 
WzH.^-^-  &c.) 

Example.  Given  I  =  100,  and  the  remaining  data  as  given  in  the 
irst  three  columns  of  the  following  table. 


Station. 

10. 

h. 

ich. 

WH. 

0 

2 

/l 

2 

1 

8< 

6 

48 

8 

2 

10.^ 

^7 

70 

56 

3 

13^ 

■^7 

91 

70 

4 

9 

"^4 

36 

52 

247 
209 
186 


186 


6)642 


10700. 


The  fourth  column  contains  the  products  of  the  several  widths  by 
the  corresponding  heights,  and  the  next  column  the  products  of  those 
widths  and  heights  touched  by  diagonals.  The  sum  of  the  products 
in  the  fourth  column  is  247,  the  sum  of  all  but  the  first  and  the  last  is 
209,  and  the  sum  of  the  products  in  the  fiftli  column  is  186.  These 
three  sums  are  added  together,  multiplied  by  100,  and  divided  by  6, 
according  to  the  formula.  This  gives  the  solidity  of  the  four  sections 
=  10700  cubic  feet. 

126.  When  the  excavation  docs  not  begin  on  a  line  at  right  angles 
lo  the  centre  line,  intermediate  stations  are  taken  where  the  excava- 
tkn  b'^gins  on  each  side  of  tlie  road-bed,  and  the  section  may  be  calcu- 


I  Ob 


EARTH-WORK. 


[ated  as  a  pyramid,  having  its  A'ertex  at  the  first  of  these  points,  and 
for  its  base  the  cross-section  at  the  second.  The  preceding  method 
gives  the  same  result,  since  w  and  h  in  this  case  become  0,  and  reduce 
;he  foraiula  to  S  ^^  i  I  w^  h^ .  The  same  remarks  apply  to  the  end  of 
an  excavation. 

C.     Grou7id  very  Irregular, 

127.  Prol>l€*m.     To  find  the  solidity  of  a  section^  when  the  ground 
is  very  irregular. 


Fig.  54. 


^ution.  Let  A  HE  FE  -  Ar  CD  Bi  F^  Ei  (fig.  54)  represent  one 
side  of  a  section,  the  surface  of  -which  is  too  irregular  to  be  divided 
into  two  planes.  Suppose,  for  instance,  that  the  ground  changes  at 
H^  C,  and  Z),  making  it  necessary  to  divide  the  surface  into  five  trian- 
gles running  from  station  to  station.*  Let  heights  be  taken  at  /7,  C, 
and  Z),  and  let  the  distances  out  of  these  points  be  measured.  If  now 
we  suppose  the  earth  to  be  excavated  vertically  downward  through 
the  side  line  B  B^  to  the  plane  of  the  road-bed,  we  may  form  as  many 
vertical  triangular  prisms  as  tliere  are  triangles  on  the  surface  This 
iviM  be  made  evident  by  drawing  vertical  planes   through  the  sides 


*  It  will  often  be  necessary  to  introduce  intermediate  stations,  in  order  to  make 
*he  subdivision  into  triangles  more  conveniently  and  accurately. 


GROUND    VERY    IRREGULAR.  109 

A  C,  H  C\  FID,  and  HB^  .     Then  the  solidify  of  the  kiJf-section  will  be 
equal  to  the  sinn  of  these  prisms,  minus  the   triangular  mass   BFG- 

BiFi  Gi  . 

The  horizontal  section  of  tlic  prisms  may  be  found  from  the  distan- 
ces out  and  the  length  of  the  section,  and  the  vertical  edges  or  heights 
are  all  known.    Hence  tl>e  solidities  of  these  prisms  may  be  calculated 

by  §  115. 

To  find  the  solidity  of  the  portion  BFG-B^  F^  Gx  ,  which  is  to 
be  deducted,  rci)resent  the  sloi)e  of  the  sides  by  s  {^  102),  the  heights 
at  B  and  B^  by  h  and  h^  ,  and  the  length  of  the  section  by  I  Then 
we  have  F  G  ^  s  /t,  and  Fi  Gi  =  shi.  Moreover,  tlie  area  of  B  F  G 
-  j s  /r,  and  that  of  B^F^G^^^s h^^.  Now  as  the  triangles  B  F  G 
and  L'l  F,  Gi  are  similar,  the  mass  required  is  the  frustum  of  a  pyra- 
mid, and  the  mean  area  is  yj  s /t^  x  i  s /'i^  =  3  '^^  ^'  ^'i  •  '^'^^^" 
(Tab.  X  53)  the  solidity  is  B F  G -  B^  F^  G^^  U s  (//-'  +  h^^  +  h h^). 

Example.  Given  Z  =  50,  6  =18,  s  =  i  ,  the  heights  at  .1,  //,  and  B 
respectively  4,  7,  and  6,  the  distimces  yl  i/  =  9  and  HB  =  9,  the 
heights  at  A^ ,  C,  D,  and  B^  respectively  C,  7,  9,  and  8,  and  the  distan- 
ces ^li  C  =4,  CD  =^  5,  and  Z)/ii  =  12  Then  the  horizontal  sec- 
tion of  the  first  prism  adjoining  the  centre  line  is  ^  /  X  A^C,  since  the 
distance  ^i  C  is  measured  horizontally  ;  and  the  mean  of  the  three 
heighta  is  ^4  +  6  +  7)  =  ^  X  17.  The  solidity  of  this  prism  is 
therefore  ^  /  X  ^li  C  X  ^  X  17  =  b  ^  X  4  X  17,  that  is,  equal  to  \l 
multiplied  by  the  base  of  the  triangle  and  by  the  sum  of  the  heights. 
In  this  way  we  should  find  for  the  solidity  of  the  five  prisms 

1/(4  X   17  +  9  X  18  +  5  X  23+  12  X  24  +  9  X  21)=  1/  X  822. 

For  the  frustum  to  be  deducted,  we  have 

^/  X  1(62  +  8^  +  6X8)  =U  X  222. 
Hence  the  solidity  of  the  half-section  is 

\l  (822  —  222)  =  g  X  50  X  600  =  5000  cubic  feet. 

128.  Let  us  now  examine  the  usual  method  of  calculating  excava- 
tun,  when  the  cross-section  of  the  ground  is  not  level.  This  method 
consists,  first,  in  finding  the  area  of  a  cross-seetion  at  each  end  of  the 
mass  ;  secondly,  in  finding  the  height  of  a  section,  level  at  the  top, 
equivalent  in  area  to  each  of  these  end  sections  ;  thirdly,  in  finding 
from  the  average  of  these  two  heights  the  middle  area  of  the  mass ; 


110  EARTH-WORK. 

and,  lastly,  in  applying  the  prismoidal  formula  to  find  the  contents 
The  heights  of  the  equivalent  sections  level  at  the  top  may  be  found 
approximately  by  Trautwine's  Diagrams,*  or  exactly  by  the  following 
method.  Let  A  represent  the  area  of  an  irregular  cross-section,  6  the 
width  of  the  road-bed,  and  s  the  slope  of  the  sides.  Let  x  be  the  re- 
quired height  of  an  equivalent  section  level  at  the  top.  The  bottom 
of  the  equivalent  section  will  be  b,  the  top  6  -f  2  s  ar,  and  the  area  will 
be  the  sum  of  the  top  and  bottom  lines  multiplied  by  half  the  height  o 
^.r  (2  6  +  2st)  =  s  X-  -\-  b  X.  But  this  area  is  to  be  equal  to  A 
Therefore,  s  x-  -\-  b  x  -^  A,  and  from  this  equation  the  value  of  a:  may 
be  found  in  any  given  case. 

According  to  this  method,  the  contents  of  the  section  already  calcu- 
lated in  §  122  will  be  found  thus.  Calculating  the  end  areas,  we  find 
the  first  end  area  to  be  387  and  the  second  to  be  240.  Then  as  s  is 
here  i  and  6=18,  the  equations  for  finding  the  heights  of  the  equiva- 
lent end  sections  will  be  ia:^  +  18x  =  387,  and  lx^-\-  ISx  =  240 
Solving  these  equations,  we  have  for  the  height  at  the  first  station 
x  =  11.146,  and  at  the  second,  x  =  S.  The  middle  area  will,  there- 
fore, have  the  height  ^  (11.146  +  8)  =  9.573,  and  from  this  height  the 
middle  area  is  found  to  be  309.78.  Then  by  the  prismoidal  formula 
(t  113)  the  solidity  will  heS^l  X  100  (387  +  240  +  4  X  309.78) 
—  31102  cubic  feet. 

But  the  true  solidity  of  this  section  was  found  to  be  32820  cubic 
feet,  a  difference  of  1718  feet.  The  error,  of  course,  is  not  in  the  pris- 
moidal formula,  but  in  assuming  that,  if  the  earth  were  levelled  at  the 
ends  to  the  height  of  the  equivalent  end  sections,  the  intervening  earth 
might  be  so  disposed  as  to  form  a  plane  between  these  level  ends,  thus 
reducing  the  mass  to  a  prismoid.  This  supposition,  however,  may 
sometimes  be  very  far  from  correct,  as  has  just  been  shown.  If  the 
diagonal  on  the  right-hand  side  in  this  example  were  reversed,  that  if 
if  the  dividing  line  were  formed  by  a  depression,  the  true  solidit} 
found  by  §  122  would  be  29600  feet ;  whereas  the  method  by  equiva- 
lent sections  would  give  the  sam.e  contents  as  before,  or  1502  feet  too 
much. 

D.     Correction  in  Excavation  on  Curves 
129.  In  excavations  on  curves  the  ends  of  a  section  are  not  parallel 


*  A  New  Method  of  Calculating  the  Cubic  Contents  of  Excavations  and  Embank 
ments  by  the  aid  of  Diagrams.     By  John  C.  Trautwine 


CORRECTION    IN    EXCAVATION    ON    CURVES. 


IIJ 


to  each  other,  but  converge  towards  the  centre  of  the  curve.  A  section 
between  two  stations  100  feet  apart  on  the  centre  line  will,  therefore, 
measure  less  than  100  feet  on  the  side  nearest  to  the  centre  of  the 
curve,  and  more  than  100  feet  on  the  side  farthest  from  that  centre. 
Now  in  calculating  the  contents  of  an  excavation,  it  is  assumed  thai 
the  ends  of  a  section  are  parallel,  both  being  perpendicular  to  the  chord 
of  the  curve.     Thus,  let  figure  55  represent  the  plan  of  two  sections  ol 


Fig.  55. 


an  excavation,  EF G  being  the  centre  line,  AL  and  Cil/the  extreme 
side  lines,  and  0  the  centre  of  the  curve.  Then  the  calculation  of  tlie 
Qrst  section  would  include  all  between  the  lines  .4 1  Ci  and  B^Di\ 
^-hile  the  true  section  lies  between  A  C  and  B  D.  In  like  manner,  the 
calculation  of  the  second  section  would  include  all  between  HK  and 
NP ,  while  the  true  section  lies  between  BD  and  L  M.  It  is  evident, 
therefore,  that  at  each  station  on  the  curve,  as  at  jP,  the  calculation  is 
too  great  by  the  wedge-shaped  mass  represented  hy  KFD^,  and  too 


Fig.  56 


^n 


■mull  by  the  mass  represented  by  BiFB      These  masses  balance 


112  EARTH-WORK. 

each  other,  when  the  distances  out  on  each  side  of  the  centre  line  are 
equal,  that  is.  when  the  cross-section  may  be  represented  hy  AD F RE 
(fig.  56).  But  if  the  excavation  is  on  the  side  of  a  hill,  so  that  the 
distances  out  differ  very  much,  and  the  cross-section  is  of  the  shape 
AD  FEE,  the  difference  of  the  wedge-shaped  masses  may  require 
consideration. 

130.  Problem.  Given  the  centre  height  c,  the  greatest  side  height  h, 
the  least  side  height  h',  the  greatest  distance  out  d,  the  least  distance  out  d', 
and  the  ividlh  of  the  road-bed  b,  to  find  the  correction  in  excavation  C,  at 
any  station  on  a  curve  of  radius  R  or  defection  angle  D. 

Solution.  The  correction,  from  what  has  been  said  above,  is  a  trian- 
gular prism  of  which  B  FR  (fig.  56)  is  a  cross-section.  The  height  of 
this  prism  at  B  (fig.  55)  is  Bi  H,  the  height  at  A'  is  R^  S,  and  the  height 
at  F  is  0.  Bi  11  and  R^  S,  being  veiy  short,  are  here  considered 
straight  lines.  Now  we  have  the  cross-secticn  B  FR  =  FB  E  G  — 
Fr'^EG  =  i^cd  +  ibh)  -  iUd'  +  ibh')  =  hc{d  -  d<)  -f 
ih{h  —  h').  To  find  the  height  Bi  H,  we  have  the  angle  B  F 11  = 
B  FBi  =  D,  and  therefore  Bi  H  =  2  HF  sin.  D  =  2d  s\n.  D.  In 
like  manner,  R^  S  =  KD^  =  2KF  sin.  D  =^  2d'  sin.  D.  Then 
since  the  height  at  Fis  0,  one  third  of  the  sum  of  the  heights  of  the 
prism  will  be  f  (d  +  fZ')sin.  D,  and  the  correction,  or  the  solidity  ol 
the  prism,  will  be  (§  115) 

^     C=[hc{d-  d')  +  ib{h-h')]  X  f(fZ-fcZ')sin.  Z). 

When  R  is  given,  iind  not  D,  substitute  for  sin.  D  its  value  (§9) 

50 
Bin.  D  =^  jf  .     The  correction  then  becomes 

^      C^[U{d-d')-^-\bih-h')]x'^^^^±^. 

This  correction  is  to  be  added,  when  the  highest  ground  is  on  the 
convex  side  of  the  curve,  and  subtracted,  when  the  highest  ground  is  on 
the  concave  side.  At  a  tangent  point,  it  is  evident,  from  figure  55,  that 
the  correction  will  be  just  half  of  that  given  above. 

Ercanple.  Given  c  =  28,  h  ^  40,  h<  =  16,  f?  =  74,  d'  =  38,  b  =  28, 
and  R.  =  1400,  to  find  C.    Here  the  area  of  the  cross-section  BFR  -= 

-  (7-i  —  38)  4-  -  (40  —  16)  =  672,  and  one  third  of  the  sum  of  the 

.     100(74  +  38)        8  ^        fi7o  V  -  « 

heights  of  the  prism  is      3  ^  ^^qq      -=  3  •      Hence  C  =  672  X  3  « 

•  792  cubic  feet. 


CORRECTION    IJ\    EXCAVATION    ON    CURVES.  113 

131.  When  the  section  is  partly  in  excavation  and  partly  in  em- 
bankment, the  cross-section  of  the  excavation  is  a  triangle  lying 
tvlioUy  on  one  side  of  the  centre  line,  or  partly  on  one  side  and  partlj 
on  the  otlier.  The  surface  of  the  ground,  instead  of  extending  from 
B  to  D  (fig.  56),  will  extend  from  B  to  a.  point  between  G  and  E,  or 
to  a  point  between  A  and  G.  In  the  first  case,  the  correction  will  be 
a  triangular  prism  lying  between  the  lines  B^  /'and  fl F  (fig.  55),  but 
not  extending  below  the  point  F.  In  the  second  case,  the  excavation 
extends  below  F,  and  the  correction,  as  in  §  129,  is  the  difference  be- 
tween the  masses  above  and  below  F.  This  difference  may  be  ob- 
tained in  a  very  simple  manner,  by  regarding  the  mass  on  both  sides 
of  i^as  one  triangular  prism  the  bases  of  which  intersect  on  the  line 
G  F  (fig.  56),  in  whicli  case  the  height  of  the  prism  at  the  edge  be- 
low /'""must  be  considered  to  be  ininus,  since  the  direction  of  this  edge, 
referred  to  either  of  the  bases,  is  contrary  to  that  of  the  two  others. 
The  solidity  of  this  prism  will  then  be  the  difference  required. 

132.  Prol>8eill.  Given  the  width  of  the  excavation  at  the  road-bed 
w,  the  ividth  of  the  road-bed  6,  the  distance  out  d,  and  the  side  height  A,  to 
find  the  correction  in  excavation  C,  at  any  station  on  a  curve  of  radius  R 
or  deflection  angle  D,  when  the  section  is  partly  in  excavation  and  partly  in 
embanlcinent. 

Solution.  When  the  excavation  lies  wholly  on  one  side  of  the  centre 
line,  the  correction  is  a  triangular  prism  having  for  its  cross-section 
the  cross-section  of  the  excavation.  Its  area  is,  therefore,  ^  iv  h.  The 
licight  of  this  prism  at  B  (fig.  56)  is  (§  130)  B^  IT  =  2  H F s\r\.  D  = 
2  d  sin.  D.  In  a  similar  manner,  the  height  at  E  will  be  2  G  E  sin.  D 
=  b  sin.  Z>,  and  at  the  point  intermediate  between  G  and  j5J,  the  dis- 
tance of  which  from  the  centre  line  is  ^t  —  ly,  the  height  will  be 
2  {^b  — 16')  sin.  D  =  (b  —  2  iv)  sin.  D.  Hence,  the  correction,  or  the  solid- 
ity of  the  prism,  will  he  {^  115)  C  =  ^whxh  {2d-i-b-{-b  —  2iv)  sin.  Z) 
-=  ^loh  X  i  {d  -\-  b  —  lo)  sin.  D. 

When  the  excavation  lies  on  both  sides  of  the  centre  line,  the  cor- 
rection, from  what  has  been  said  above,  is  a  triangular  prism  having 
also  for  its  cross-section  the  cross-section  of  the  excaration.  Its  area 
will,  therefore,  be  ^ivh.  The  height  of  this  prism  at  Bis  also  2dsin.D, 
and  the  height  at  E,  b  sin.  D ;  but  at  the  point  intermediate  between  A 
and  G.  the  distance  of  which  from  the  centre  line  \s  w  —  ^b,  the  height 
will  be  2  (iv  —  ^b)  sin.  Z)  =  (2  lo  —  b)  sin.  D.  As  this  height  is  to 
be  considered  minus,  it  must  be  subtracted  from  the  others,  and  the 
coriection  required  will  be  C=^wkxhi2d-\-b  —  2w-\-b)  sin.  D 


114  EAETH-WORK. 

^  ^wh  X  I  (^  +  t  —  't')  sin.  D.  Hence,  in  all  cases,  when  the  sec 
tion  is  partly  in  excavation  and  partly  in  embankment,  we  have  the 
formula 

1^-  C=^'u;hX  ^  {d-\-b—  iv)  sin.  D. 

When  R  is  given,  and  not  D,  substitute  for  sin.  D  its  value  (§  9) 

50 
sin.  D  =  -^  .    T^e  correction  then  becomes 

This  correction  is  to  be  added,  when  the  highest  ground  is  on  the 
convex  side  of  the  curve,  and  subtracted  when  the  highest  ground  is  on 
the  concave  side.  At  a  tangent  point  the  correction  will  be  just  half 
of  that  given  above. 

Example.     Given  if;  =  17,  6  =  30,  c?  =  51,  A  =  24,  and  22  =  1600, 

to  find  C.    Here  the  area  of  the  cross-section  is  ^wh  =  \7  %  12  = 

.    If0(d+b—w) 
204-.  and  one  third  of  the  sum  of  the  heights  of  the  prism  is  g^j 

..^  ^^"^^^^^  =  l     Hence  C  =  204  X  |  =  272  cubic  feet. 

1.33.  The  preceding  corrections  (§130  and  ^32)  suppose  the  length 
of  the  sections  to  be  100  feet.  If  the  sections  are  shorter,  the  angle 
B  FH  (fig.  55)  may  be  regarded  as  the  same  part  of  D  that  FG  is  ol 
100  feet,  and  Sj  FB  as  the  same  part  of  D  that  jEJFis  of  100  feet 
The  true  correction  may  then  be  taken  as  the  same  part  of  C  that  the 
mm  of  the  lengths  of  the  two  adjoining  sections  is  of  200  feet. 


TABLE    I. 


UADII,    ORDINATES,    DEFLECTIONS, 


AND 


ORDINATES  FOR  CURVING  RAILS. 

Jroraiiila  for  Radii,  ^  10  ;   for  Ordinates,  §  25 ;  for  Dcflectlong,  $  1*J 

for  CuiTiug  Riiils,  §  29. 


lib 

TABLE 

I.       RADII 

,    ORDINATES,     DEFLECTIONS, 

Degree. 

Radu. 

Ordinates. 

Tangent 
Deflec- 

Chord 
Deflec- 

Ordinates for 
Rails. 

12^ 

25. 

37i. 

50. 

tion. 

tion. 

18. 

20. 

O      ( 

0    5 

6S754.94 

.008 

.014 

.017 

.018 

.073 

.145 

1 
.001 

.001 

10 

34377.48 

.016 

.027 

.034 

.036 

.145 

.291 

.001 

.001 

15 

22918  33 

.024 

.041 

.051 

.055 

.218 

.436 

.002 

.002 

20 

171SS.76 

.032 

.055 

,063 

.073 

.291 

.582 

.002 

.003 

251 

13751.02 

mo 

.063 

.085 

.091 

.364 

.727 

.003 

.004 

30 

11459.19 

.013 

.032 

.102 

.109 

.4.36 

.873 

.004 

004 

35 

9322. 1-? 

.056 

.095 

119 

.127 

.509 

1.013 

.004 

.005 

40 

8594.41 

.064 

.109 

.136 

.145 

.532 

1.164 

.005 

.006    1 

45 

7639.49 

.072 

.123! 

.153 

.164 

.654 

1.309 

.005 

.007     1 

50 

6375.55 

.080 

.136 

.170 

.132 

.727 

1.454 

.006 

.007 

55 

6250.51 

.037 

.150 

.187 

.200 

.800 

1.600 

.006 

.008 

1     0 

5729.65 

.095 

.164 

.205 

.218 

,873 

1.745 

.007 

.009 

5 

523S.92 

103 

.177 

.222 

.236 

,945 

1.891 

.008 

.009 

10 

4911.15 

.111 

.191 

.239 

.255 

1,018 

2.036 

.008 

.010 

15 

45S3.75 

.119 

.205 

.256 

.273 

1.091 

2.182 

.009 

.011 

20 

4297. 2S 

.127 

.218 

.273 

.291 

1.164 

2.327 

.009 

.012 

25 

4044.51 

.135 

.232 

.290 

.309 

1 .236 

2.472 

.010 

.012 

30 

33 19.  S3 

.143 

.245 

.307 

.327 

1.3(19 

2.613 

.011 

.013 

35 

36  IS.  SO 

.151 

.259 

..324 

.345 

1  332 

2.763 

.011 

.014 

)          4C 

3437.87 

.159 

.273 

..341 

.364 

1.454 

2.909 

.012 

.015 

45 

3274.17 

.167 

.236 

.358 

.332 

1.527 

3.054 

.012 

.015 

50 

3125.36 

.175 

.300 

.375 

.400 

1.600 

3.200 

.013 

.016 

55 

2939.43 

.133 

.314 

.392 

.418 

1.673 

3.345 

.014 

017 

9    0 

2S64.93 

.191 

.327 

.409 

.436 

1.745 

3.490 

.014 

.017 

5 

2750.35 

.199 

.341 

.426 

.455 

1.818 

3.636 

.015 

.013 

in 

2644.53 

.207 

.355 

.443 

.473 

1.391 

3.781 

.015 

.019 

15 

2546.64 

.215 

.363 

.460 

.491 

1.963 

3.927 

.016 

.020 

20 

2455.70 

.223 

.3c2 

.477 

.509 

2.036 

4.072 

.016 

.020 

25 

2371.04 

.231 

.395 

.494 

.527 

2.109 

4.218 

.017 

.021 

30 

2292.01 

.239 

.409 

.511 

.545 

2.1S1 

4.363 

.018 

.022 

35 

2213.09 

.247 

.423 

..528 

.564 

2.251 

4.503 

.018 

.023 

40 

2143.79 

.255 

.436 

..545 

.582 

2.327 

4.654 

.019 

.023 

45 

2033.6S 

.263 

.450 

.562 

.600 

2.400 

4.799 

.019 

.024 

50 

2022.41 

.270 

.464 

.530 

.613 

2.472 

4.945 

.020 

.025 

55 

1664  64 

.278 

.477 

.597 

.636 

2.545 

5.090 

.021 

.025 

3    0 

1910.03 

.286 

.491 

.614 

.655 

2.6  IS 

5.235 

.021 

.026 

5 

1358.47 

•294 

.505 

.631 

.673 

2.690 

5..381 

.022 

.027 

10 

1309.57 

.302 

.518 

.643 

.691 

2.763 

5.526 

.022 

.028 

15 

1763  13 

.310 

..532 

.665 

.709 

2.336 

5.672 

.023 

!     .023 

20 

1719.12 

.318 

.545 

.682 

.727 

2.908 

5.817 

.024 

1     .029 

25 

1677.20 

.326 

.559 

.699 

.745 

2.9S1 

5.962 

.024 

1     .030 

30 

1637.28 

.3-34 

.573 

.716 

.764 

3.054 

6.108 

.025 

j     .031 

35 

1599.21 

.342 

.536 

.733 

.782 

3.127 

6.2.53 

.025 

.031 

40 

1 562.SS 

.3.50 

.600 

.750 

.800 

3.199 

6.398 

.026 

,     .032 

45 

1523.16 

.353 

.614 

.767 

.818 

3.272 

6.544 

.027 

I     .033 

50 

1494.95 

.366 

.627 

.784 

.8.36 

3.345 

6.639 

,027 

j     .033 

55 

1463  16 

.374 

.641 

.801 

.855 

3  417 

6,835 

.028 

.034 

4    0 

(432.69 

.332 

.655 

.818 

.873 

3.490 

6.930 

.028 

.035 

5 

1403  46 

.390 

.663 

.835 

.891 

3.563 

7.125 

.029 

.036 

10 

1375.40 

.398 

.632 

.852 

.909 

3.635 

7.271 

.029 

;      .036 

15 

1343.45 

.406 

.695 

.869 

.927 

3.703 

7.416 

.030 

.037 

20 

1.3.22.  .53 

.414 

.709 

.836 

.945 

3.731 

7..561 

.031 

.033 

25 

1297.53 

.422 

,723 

.903 

,964 

3.3.53 

7.707 

.031 

!     .039 

30 

1273.57 

.430 

.736 

.921 

.932 

3.926 

7.352 

.032 

.039 

35 

12.50.42 

.438 

.750 

.933 

1.000 

3.999 

7.997 

.032 

.040 

40 

1223.11 

.446 

.764 

.955 

1.018 

4.071 

8.143 

.033 

.041 

45 

1206.57 

.454 

.777 

.972 

1.036 

4.144 

8.2.8S 

.034 

.041 

50 

1185.78 

.462 

.791 

.939 

1.055 

4.217 

8,4:33 

.034 

.042 

55 

1165.70 

.469 

.805 

1.006 

1.073 

4.239 

8.579 

.035 

.043 

5    0 

1146.23 

.477 

.818 

1.023 

1.091 

4.362 

8.724 

.035 

.044 

AND^ORDINATES    FOR    CURVING    RATI  S. 


117 


Degree. 


Radii. 


o    / 
5    5 

10 
15! 

201 
25 
30 
35 
40 
45 
50 
55 

6     0 
5 

10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

7    0 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 


8  0 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

9  o! 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 


Ordinates. 


12i. 


i  127.50 

1  [09.33 

1091.73 

1U74.68 

1058.16 

1042.14 

1026.60 

1011.51 
996.87 
982.61 
968.81 

955  37 
912.29 
929  57 
917.19 
905.13 
893.39 
SSI.  95 
870.79 
859.92 
849.32 
838.97 
828.88 

819.02 
809.40 
800.00 
790.81 
781. S4 
773.07 
764.49 
756.10 
747.89 
739.86 
732.01 
724.31 

716.78 
709.40 
702.18 
695.09 
688.16 
681.35 
674.69 
66S.15 
661.74 
655.45 
619.27 
643.22 

637.27 

631.44 

625  71 

620.09 

614. r,6 

609.14 

603.80 

598.57 

593.42 

588.36 

583.38 

578.49 


25. 


.4; 

.493 

501 

.509 

.517 

.525 

.533 

.541 

.549 

.557 

.565 


37*. 


60. 


10    0       573.69 


.581 

.589 

..597 

.605 

.613 

.621 

.629 

.637 

.645 

.653 

.66 

.669 
.677 
.685 
.693 
,701 
.709 
.717 
.725 
.733 
.740 
.748 
.756 

.764 

.772 
.780 
.788 
.796 
.804 
.812 
.820 
.828 
.836 
.844 
.852 

.860 
.868 
.876 
.884 
.892 
.900 
.908 
.916 
.924 
,932 
.940 
.948 

.956 


.832 

.846 1 
,859 
.873! 
.887 
.900 
,914 
.928 
,941 
.955 
.96^ 

,982 

,996 
1,009 
1,023 
1.037 
1,050 
1.061 
1.078 
1.091 
1.105 
1.118 
1.132 

1.146 
1.1.59 
1.173 
1.187 
1.200 
1.214 
1.228 
1.242 
1.255 
1.269 
1.283 
1.296 

1.310 
1,324 
1,337 
1,351 
1.365 
1.378 
1.392 
1,406 
1.419 
1.433 
1.447 
1,460 


1.040 
1,057 
1,074 
1.091 
1.108 
1.125 
1.142 
1.1.59 
1.176 
1.193 
1.210 

1.228 
1.24.''i 
1.262 
1.279 
1.296 
1.313 
1 .330 
1.347 
1.364 
1.381 
1.398 
1.415 

1.432 
1.449 
1.466 
1.483 
1.501 
1.517 
1.535 
1.552 
1.569 
1.586 
1.603 
1.620 

1.637 
1.6.54 
1.671 
1.688 
1.705 
1.722 
1.739 
1.757 
1.774 
1.791 
1.808 
1.825 


Tangent 
Petiec- 
.tion. 


1.109 
1.127 
1.146 
1.164 
1.182 
1  200 
1.218 
1.237 
1.255 
1 .273 
1.291 

1.309 
1.327 
1.346 
1.364 
1.382 
1.400 
1.418 
1.437 
1 .455 
1.473 
1.491 
1.510 

1 .528 
1.546 
1.564 
1 .582 
1 .600 
1.619 
1.637 
1,655 
1,673 
1.691 
1.710 
1.728 

1.746 
1.764 
1.782 
1.801 
1.819 
1.8.37 
1.8.55 
1.873 
1.892 
1.910 
1.928 
1.940 


1.474 
1.488 
1.501 
1.515 
1,529 
I, .54  2 
1,.556 
1,570 
1.583 
1.597 
1.611 
1.624 

1 .638 


1.842 
1.859 
1.876 
1.893 
1.910 
1.927 
1.944 
1.961 
1.979 
1.996 
2.013 
2.030 

2.047 


1.965 
1.983 
2.001 
2.019 
2.037 
2.056 
2.074 
2.092 
2.110 
2.128 
2.147 
2.165 

2.183 


Chord 

Ufllcc- 

tion. 


Oldir.ates  fen- 
Rails. 


4.435 
4.507 
4.580 
4.653 
4.725 
4.798 
4.870 
4.943 
5.016 
-  5.088 
5.161 

5.234 

5.306 

5..379 

5.451 

5.524 

5..597 

5.669 

5.742 

5.814 

5.8S7 

5.960 

6.032 

6.105 
6.177 
6.250 
6.323 
6.395 
6.468 
6.540 
6.613 
6.685 
6,758 
6.831 
6.903 

6.976 
7.048 
7.121 
7.193 
7.266 
7,338 
7.411 
7.483 
7,556 
7,628 
7.701 
7.773 

.7.846 
7  918 
7.991 
8.063 
8.136 
8.208 
8.281 
8.353 
8.426 
8.49S 
8.571 
8.643 

8.716 


8.869 

9.014 

9.160 

9.305 

9.450 

9.596 

9.741 

9.8.-^6 

10.031 

10.177 

10.322 

10.467 
10.612 
10.758 
10.903 
11.048 
11.193 
11.339 
11.484 

11.774 
11  919 
r2.u65 


12.210 

12.355 

12.500 

12.645 

12.790  j 

12.936 

13.081 

13.226 

13.371 

13.516 

13.661 

13.806 

13.951 

14.096 

14.241 

14.387 

14.532 

14.677 

14.822 

14.967 

15.112 

15.257 

15.402 

15.547 

15.692 
15.837 
15.9S2 
16.127 
16.272 
16.417 
16.562 
16.707 
16.852 
16.996 
17.141 
17.286 

17.431 


18. 

.036 
.037 
.037 
.038 
.038 
.039 
.039 
.n4() 
.041 

.(!41 

.0-/2 

.042 

.043 

.044 

.044 

.04 

.04 

.046 

.047 

.047 

.(48 

.048 

.049 

.049 

.050 

.051 

.051] 

.052 

,052 

,053 

.054 

,054 

,055 

.055 

.056 


20. 


'I 


.057 

.057 

.058 

.058 

.059 

.0591 

.060 

.061 

.061 

.062 

.062 

.063 

.064 

.064 

.065 

.065 

.066 

.0661 

.0671 

.068 

.0681 

.069 

.069 

,070 


.044 
,045 
,046 
,047 
,047 
.048 
.049 
.049 
.050 
,051 
,052 

,052 
.053 
.054 
.055 

.055 
.056 
.057 
.057 
.058 
.059 
.060 
.060 

061 

.062 

.063 

.063 

.064 

.065 

.065 

.066 

.067 

.068 

.068 

.069 

070 
,070 
,071 
,072 
.073 
.073 
.074 
.075 
.076 
.076 
.077 
.078 

.078 
.073 
.080 
,081 
.081 
.082 
.083 
.084 
.084 
.085 
.0>6 
,086 


.071      .087 


118 


TABLE    I.       RADII,    ORDINATES,    DEFLECTIO.NS,    i^C. 


r 
Degree. 

Radii. 

Ordinates. 

Tangent 
Deflec- 
tion. 

'  Chord 
Deflec- 
tion. 

Ordinates  for 
Rails. 

12^. 

25. 

37*. 

50. 

18. 

20. 

o    / 
lU  IJ 

564.31 

.97-2 

1.665    2.031 

2.219 

8.S6C 

17.721 

.072 

.039 

2] 

555. '23 

.933    1.693    2.115    2.2.56 

9.OO0 

13.01 1 

.073 

.090 

33 

546.44 

l.OM    l.720l  2.149;  2.292 

9.150 

13.300 

.074 

.092 

40 

537.92 

1.020    1.743 

2.131 

2.329 

9.295 

13.590 

.075 

.093 

50 

529.67 

1.036 

1.775 

2.213 

2.355 

9.440 

13.330 

.076 

.094 

11     0 

521.67 

1.052 

1.302 

2.252'  2.402 

9.535 

19.169 

.073 

.098 

10 

51.3.91 

I  mi 

1.S30 

2.236:  2.4.33 

9.729 

19.459 

.079 

.097 

20 

506.33 

1.0S4 

1 .857 

2.320 

2.475 

9.374 

19.743 

.030 

.099 

30 

499.136 

l.ldOl   1.334 

2.3>1 

2.511 

10.019 

20.0:33 

.031 

.100 

40 

491.96 

l.llG    1.912 

2.339 

2.;547 

10.164 

20.327 

.032 

.102 

50 

4S5.05 

1.132    1.9:33 

2.423 

2.531 

10.:303 

20.616 

.034 

.103 

12    0 

47S.ai 

1.143 

1.967 

2.457 

2.620 

10.453 

20.906 

.035 

.105 

10 

471.31 

1.164 

1.994 

2.491 

2.657 

IO..597 

21.195 

.036 

.106 

20 

465.46 

1.130 

2.021 

2.525 

2.693 

10.742 

21.434 

.087 

.107 

30 

459. 2S 

I.I96I  2.049 

2.560 

2.730 

10.337 

21.773 

.088 

.109 

40 

4-53.26 

1.212    2.076 

2.594 

2.766 

11.031 

22.063 

.039 

.110 

50 

447.40 

1.223    2.104 

2.623 

2.303 

11.176 

22.-352 

.091 

.112 

13    0 

441.63 

1J244    2.131 

2.662 

2.839 

11.320 

22.641 

.092 

.113 

10 

436. 12 

1.260    2.159 

2.697 

2.376 

11.465 

22.930 

.093 

.115 

20 

430.69 

1.277 

2. 1 36 

2.731 

2.912 

11.609 

23.219 

.094 

.116 

30 

425.40 

1.293 

2.213 

2.765 

2.949 

11.754 

2:3.507 

.095 

.113 

40 

420.23 

i.:3a9 

2.241 

2.799 

2.935 

11.393 

23.796 

.096 

.119 

50 

41.5.19 

1.325 

2.263 

2.3.3:3 

3.022 

12.013 

24.035 

.093 

.120 

14    0 

410.23 

1.341 

2.296'  2.363 

3.053 

12.137 

24.374 

.099 

.122 

10 

40.5.47 

1.357 

2.323 

2.902 

3.095 

12.331 

24.663 

.100 

.123 

20 

400.73 

1.373 

2.351 

2.9.361  3.131 

12.476 

24.951 

.101 

.125 

30 

396.20 

l.:3S9 

2.373 

2.970 

3.163 

12.620 

25.240 

.102 

.126 

40 

391.72 

1.405 

2.406 

3.005 

3.204 

12.761 

25.523 

.103 

.123 

50 

337.34 

1.421 

2.4.33 

3.039 

3.241 

12.903 

25.317 

.105 

.129 

15    0 

333.06 

1.4:37 

2.461 

3.073 

3.277 

1:3.0.53 

26.105 

.106 

.131 

10 

373.33 

1.4.53 

2.4  S3 

3.107 

3.314 

1:3.197 

26.394 

.107 

.1.32 

20 

374.79 

1.469 

2.515 

3.142 

3.350 

I3.:341 

26.632 

.103 

.133 

30 

370.73 

1.436 

2..543 

.3.176;  3.337 

13.435 

26.970 

.109 

.135 

40 

366.36 

1.502 

2.570 

3.210    3.423 

13.629 

27.253 

.110 

.136 

50 

363.02 

1.513 

2.593 

3.245 

3.460 

13.773 

27.547 

.112 

.133 

16    0 

3.59.26 

1..5.34 

2.625 

3.279 

3.496 

13.917 

27.335 

.113 

.139 

10 

355. 59 

1.550 

2.6.53 

3.313 

3.5:33 

14.061 

23.123 

.114 

.141 

20 

351.93 

1.566 

2.630 

3.317 

3.569 

14.205 

23.411 

.115 

.142 

30 

a43.45 

1.532 

2.703 

3.332    3.606 

14.349 

23.699 

.116 

.143 

40 

344  99 

1.593 

2.7.36 

3.416    3.643 

14.493 

23.936 

.117 

.145 

50 

ail. 60 

1.615 

2.763 

3.450 

3.679 

14.637 

29.274 

.119 

.146 

17    0 

33S.27 

1.631 

2.791 

3.435 

3.716 

14.731 

29.562 

.120 

.143 

10 

335.01 

1.617 

2.313 

3.519 

3.7.52 

14.925 

29.3-50 

.121 

.149 

20 

3:31.82 

1.663 

2.346 

3.-5.53 

3.739 

15.069 

30.137 

.122 

.151 

30 

323.63 

1.679 

2.373 

3.583 

3.325 

15.212 

30  425 

.123 

.152 

40 

32-5.60 

1.695 

2.901 

3.622 

3  362 

15.356 

30.712 

124 

.154 

50 

322.59 

1.711 

2.923 

3.656 

3.393 

15.500 

31.000 

.126 

.155 

18    0 

319.62 

1.723 

2.956 

3.691 

3.935 

15.643 

31.237 

.127 

.1-56 

10 

316.71 

1.744 

2.933 

3.725 

3.972 

1-5.737 

31.574 

.123 

.153 

20 

313.36 

1.760 

3.011 

3.7.59 

4.003 

15.931 

31.361 

.129 

.159 

30 

311.06 

1.776 

3.0.39 

3.794 

4.045 

16.074 

32.149 

.130 

.161 

40 

303.30 

1.792:  3.066  i 

3.523 

4.081 

16.213 

32.436 

.131 

.162 

50 

305.60 

1.309 

3.094 

3.362 

4.113 

16.361 

32.723 

.133 

.164 

19    0 

302.94 

1.325 

3.121 

3.397 

4.1.55 

16.-505 

33.010 

.134 

.165 

10 

3D0..33 

1.341 

3.149 

3.931 

4.191 

16.643 

33.296 

.135 

.166 

20 

297.77 

1.357 

3.177 

3.965 

4.223 

16.792 

a3.533 

.1-36 

.163 

30 

295.25 

1.373 

3.204 

4.000 

4.265 

16.9-35 

33.870 

.137 

.169 

40 

292.77 

1.390 

3.232 

4.034 

4.301 

17.073 

a4.157 

.1-33 

.171 

50 

290.33 

1.906    3.2.59  j 

4.069 

4.333 

17.222 

34.443 

.140 

.172 

20    0 

237.91 

1.922;  3.2371 

4.103 

4.374 

17.365 

34.730 

.141 

.174 

TABLE    II.       LONG    CHORDS. 


119 


TABLE    II. 
LONG   CHORDS.    §  69. 


Degree  of 
Civrve. 

2  Stations. 

3  Stations. 

4  Stations. 

5  Stations. 

.. ., 
6  Stations.      ! 

o    t 
0  10 

200.000 

299.999 

399.993 

499.996 

599.993 

20 

199.999 

.997 

.992 

.953 

.970 

30 

.993 

.992 

.931 

.962 

.933 

40 

.997 

.936 

.966 

.932 

.832 

50 

.995 

.979 

.947 

.894 

.815 

1     0 

199.992 

299.970 

399.924 

499.S43 

599.733 

10 

.990 

.959 

.896 

.793 

.637 

20 

.956 

.946 

.865 

.729 

.526 

30 

.9S3 

.932 

.829 

.657 

.401 

40 

.979 

.915 

.789 

.577 

.260 

oO 

.974 

.893 

.744 

.483 

.105 

2    0 

199.970 

299.378 

399.695 

499.-391 

595.934        i 

10 

.964 

.857 

.643 

.255 

.7.50        1 

20 

.959 

.SM 

.5^36 

.171 

.5:50 

30 

.9.')2 

.810 

.524 

•ai9 

.336 

40 

.916 

.733 

.459 

498.913 

.106 

50 

.939 

.756 

.389 

.778 

597,662 

3    0 

199.931 

299.726 

399.315 

49S.630 

597.604 

10 

.924 

.695 

.2.37 

.474 

.331 

20 

.915 

.652 

.154 

.309 

.043 

30 

.907 

.627 

.Oft3 

.136 

596.740 

40 

.893 

.591 

398.977 

497.955 

.423 

50 

.833 

.553 

.882 

.765 

.091 

4    0 

199.S73 

299.513 

393.732 

497.566 

595.744 

10 

.563 

.471 

.679 

.360 

.353 

20 

.857 

.423 

.571 

.145 

.007 

30 

.846 

.333 

.459 

496.921 

594.617 

40 

.834 

.337 

.343 

.639 

.212 

50 

.822 

.239 

.223 

.449 

593.792 

5    0 

199.810 

299.239 

393.099 

496.200 

593.353 

10 

.797 

.157 

397.970 

495.944 

592.909 

20 

.733 

AM 

.837 

.678 

446 

30 

.770 

.079 

.709 

405 

591.963 

40 

.756 

.023 

.559 

.123 

.476 

50 

.741 

293.9&1 

.413 

494.832 

590.970 

6    0 

199.726 

293.904 

397.264 

494.5^4 

590.449 

10 

.710 

.843 

.110 

.227 

589.913 

20 

.695 

.779 

396.952 

493.912 

.364 

30 

.673 

.714 

.790 

.553 

533.300 

40 

.662 

.643 

.6-23 

257 

.221 

50 

.644 

.579 

453 

492.917 

537.623 

7    0 

199.627 

298.509 

396.278 

492.563 

537.021 

10 

.609 

433 

099 

.212 

536.400 

20 

.591 

.3&1 

395.916 

491.347 

535.765 

30 

.572 

.239 

.729 

.474 

.115 

40 

•553 

.212 

.533 

.093 

584.451 

50 

.533 

.134 

.342 

490.701 

533.773 

8    0 

.513 

293.054 

395.142 

490.306 

553.051 

120 


TABLE    III.  TABLE    IV. 


TABLE    III. 

CORRECTION  FOR   THE   EARTH'S   CURVATURE    AND 
FOR   REFRACTION.    §  105. 


D. 

J. 

D. 

d. 

D. 

d. 

D. 

d. 

303 

.002 

ISOO 

.066 

3300 

.223 

4--'00 

.472 

400 

.0ft3 

1900 

.074 

3  J  00 

.237 

4900 

.492 

500 

.005 

2CH30 

.0S2 

3^500 

.25! 

5000 

.512 

600 

.007 

2100 

.090 

36Q0 

.266 

5100 

.533 

700 

.010 

2200 

.099 

37(10 

.2S1 

5200 

,554 

800 

.013 

2300 

.lOS 

3S00 

.2i;6 

1  mUe 

.571 

900 

.017 

2400 

.113 

3900 

.312 

2     «♦ 

2.235 

1000 

.020 

2500 

.123 

4000 

.328 

3    »< 

5.142 

1100 

.025 

2600 

.139 

4100 

.345 

4    « 

9.142 

3200 

.030 

2700 

.149 

4200 

.362 

5    « 

14.284    { 

1300 

.035 

2SitO 

.161 

4300 

.370 

6    " 

20.563 

1400 

.040 

2900 

.172 

4400 

.397 

7     " 

27.996 

1500 

.046 

3000 

.1S4 

4501 

.415 

8    " 

36.566 

1600 

052 

3100 

.197 

460  J 

.434 

9    « 

46.279 

1700 

.059 

3200 

.210 

4700 

.453 

10    " 

57.135 

TABLE    IV. 

ELEVATION   OF   THE   OUTER   RAIL  ON   CURVES. 

§  110. 


Degree. 

RT  =  15 

M  =  20. 

M  =  26. 

M  =  80. 

M  =  40. 

M  =  50. 

o 

1 

.012 

.022 

034 

.049 

.088 

.137 

2 

.025 

.044 

.068 

.099 

.175 

.274 

3 

.037 

.066 

.103 

.143 

.263 

.411 

4 

.049 

.033 

137 

.197 

.351 

.543 

5 

.062 

.110 

.171 

.247 

.433 

.685 

6 

.074 

.131 

.205 

.296 

.526 

.822 

7 

.0S6 

.153 

.240 

.345 

.613 

.953 

8 

.099 

.175 

.274 

.394 

.701 

1.095 

9 

.111 

.197 

.303 

.443 

.788 

1.232 

10 

.123 

.219 

.342 

.493 

.876 

L36S 

TABLE    V.  TABLE    VI. 


121 


TABLE    V. 


FROG  ANGLES,   CHORDS,  AND   ORDINATES  FOR 

TURNOUTS. 

This  table  is  calculated  for  g  =  4.7,  d  =-  .42,  and  S  =  1°  20'.  For 
mula  for  frog  angle  F,  and  chord  B  F,  §  50 ;  for  m,  the  middle  or- 
dinate of  B  F,  §  25  ;  for  ?/i',  the  middle  ordinate  for  curving  an  18  ft 
rail,  §  29. 


R. 

imo 

F. 

BF. 

in. 

niK 

R. 

600 

F. 

BF. 

m. 

m' 

g  27  ik 

72.22 

.651 

.041 

O      1 

6  57 

48 

59.17 

.727 

.068 

975 

5  31  39 

71.53 

.655 

.012 

575 

7  6 

26 

58.16 

.7.33 

.070 

950 

5  35  44 

70.S3 

.659 

,043 

550 

7  15 

40 

57.12 

.739 

.074 

925 

5  39  59 

71.11 

.663 

,044 

525 

7  25 

33 

56.05 

,745 

,077 

900 

5  44  24 

69.3S 

.667 

,045 

500 

7  36 

10 

54.94 

.752 

,081 

875 

5  49  1 

68.64 

.671 

.046 

475 

7  47 

37 

53.79 

.758 

.085 

850 

5  53  50 

67.88 

.676 

,01S 

450 

8  0 

1 

52.61 

.765 

.090 

825 

5  53  52 

67.10 

.680 

,049 

425 

8  13 

30 

51. 3r 

.773 

.095 

ST) 

6  4  9 

66.30 

.685 

,051 

400 

S  23 

14 

50.09 

.780 

.101 

775 

6  9  41 

6."'.49 

.690 

.052 

375 

8  44 

26 

48.75 

.788 

.103 

750 

6  15  30 

64.65 

.695 

.054 

350 

9  2 

20 

47.35 

.796 

.116 

725 

6  21  37 

63.80 

.701 

.056 

325 

9  22 

16 

45.88 

.805 

.125 

700 

6  28  4 

62.92 

.705 

.058 

300 

9  44 

39 

44.34 

.814 

.135 

675 

6  34  52 

62.02 

.710 

,060 

275 

10  10 

1 

42.72 

.824 

.147 

650 

6  42  4 

61.09 

.716 

,062 

250 

10  39 

6 

41.00 

.834 

.162 

625 

L—  ...  .  . 

6  49  42 

60,14 

.721 

.065 

225 

11  12 

55 

39.16 

.845 

.180 

TABLE    VI. 

LENGTH  OF   CIRCULAR  ARCS  IN   PARTS   OF  RADIUS 


o 

1 

,01745 

32925 

19943 

1 

.00029 

08882 

08666 

// 

1 

,00000 

48481 

36811 

9. 

.03490 

65850 

39.S87 

2 

.00058 

17764 

17331 

2 

,00000 

96962 

73622 

3 

.05235 

98775 

59830 

3 

.00087 

26646 

25997 

3 

.00001 

45444 

10433 

4 

,(;69S1 

31700 

79773 

4 

.00116 

35528 

34663 

4 

.00001 

9-3925 

47244 

ri 

.03726 

64625 

997 1 6 

5 

.00145 

44410 

43329 

5 

,00002 

42406 

84055 

6 

,10471 

9755 1 

19660 

6 

.00174 

53292 

51994 

6 

.00002 

90.888 

20,867 

7 

.12217 

.30476 

39603 

7 

,00203 

62174 

60660 

7 

.00003 

39369 

57678 

8 

.13962 

63401 

59546 

8 

,00232 

71056 

69326 

8 

,00003 

87850 

94489 

9 

.15707 

96326 

79190 

J_ 

,00261 

79933 

77991 

9 

,00004 

36332 

31300 

122 


TABLE    VII.       EXPANSION    BY    HEAT. 


TABLE    VII. 


EXPANSION  BY  HEAT. 


Bodies. 

323  to  2123. 

lO. 

Authority. 

Platina, 

.0003S42 

.000004912 

Ilassler 

Gold, 

,001466 

.000003141 

(( 

Silver, 

.001909 

.000010605 

(( 

Mercury, 

.01S013 

.0001001 

(( 

Brass, 

.00189163 

.000010509 

(( 

Iron, 

.00125344 

,000006964 

(( 

^V'ater, 

.0466 

not  uniform. 

(( 

Granite, 

.00036350 

.0000(MS25 

Prof.  Bartlett. 

Marble, 

.00102024 

.00000566.3 

(( 

Sends  tone, 

.00171576 

.000009532 

u 

TABLE    VIII,        PROPERTIES    OF    MATEUIALS. 


123 


TABLE    VIII. 


PROPERTIES   OF  MATERIALS. 


The  authorities  referred  to  by  the  capital  letters  in  the  table  are :  — 


B        Barlow,  On  the  Strength  of 
Materials. 
Bevan. 

Lieut.  Brown. 
Couch. 
Franklin  Institute,  Report  on 

Steam  Boilers. 
Gordon,  Eng.  Translation  of 

Weisbach. 
Hodgkinson,  Reports  to  Brit. 
Association. 
Ha,     Hassler,  2\ibles. 


Be 
Br 
C. 
F. 


G. 


H. 


L.       Lame. 

M.  Musschenbroek,  Int.  to  Nat 
Phil. 

R.       Rennie,  Pliil.  Trans. 

Ro.     Rondelet,  Vxirt  de  Batir. 

T.       Telford. 

Ta.     Taylor,  Statistics  of  Coal. 

W.  Weisbach,  Mech.  of  Machin- 
ery and  Engineering. 

The  numbers  without  letters  ar« 
taken  from  Prof  Moseley's  En- 
gineering and  Architecture 


In  finding  the  weights,  a  cubic  foot  of  water  has,  for  convenience, 
been  taken  at  62.5  lbs. 

The  numbers  for  compression  taken  from  Hodgkinson  were  ob- 
tained by  him  from  prisms  high  enough  to  allow  the  wedge  of  rupture 
to  slide  freely  off.  He  shows  that  this  is  essential  in  experiments  on 
rompression. 

The  modulus  of  rupture  *S  is  the  breaking  weight  of  a  prism  1  in 
broad,  1  in.  deep,  and  1  in.  between  the  supports,  the  weight  being  ap- 
plied in  the  middle.     To  find  the  corresponding  breaking  weight  I^of 
a  rectangular  beam  of  any  other  size,  let  /  =  its  length,  b  =:  its  breadth, 

2  b  d'i 
and  d  =  its  depth,  all  in  inches.     Then  W  =  -or    X  'S. 

The  numbers  in  the  last  three  columns  express  absolute  strength 
For  safety,  a  certain  proportion  only  of  these  numbers  is  taken.     The 
divisors  for  wood  may  be  from  6  to  10,  for  metal  from  3  to  6,  for  stone 
10,  and  for  ropes  3. 

When  double  numbers  are  used  in  the  column  headed  "  Crushing 
Force  per  Square  Inch  in  lbs.,"  the  first  applies  to  specimens  moder- 
ately dry,  the  second  to  specimens  turned  and  kept  dry  in  a  warm 
place  two  months  longer.  In  the  case  of  American  Birch,  Elm,  and 
Teak,  the  numbers  apply  to  seasoned  specimens. 


134 


T.ABLE    VIII. 


'ROPERTIES    OF    MATERIALS. 


Materials. 


it 


Metals. 

Ccppe.',  oapt,  .  .  , 
rLllcd,  .  , 
r'iie-firawTi, 

GoH, 

Iron,  cast, 
Canou  Xo.  2,  cold 
"  •'      hot 

Devon  No.  ?,  ccld 
'      hoi 
Butlery  yo.  1,  ci  la" 
''  "     hoi 

Iron,  wroughS, 
Encjlish  bar, 
Welsh      " 
Swedish   "  .     . 

Lancaster  v'o  ,  / 
Tenness'ie 
Missouri 
Iron  wire, 
Enslish,     a'an 
rhmipsb'g,  ra. 


blast, 

(( 

u 

kC 

u 


Lead,  cast,   .     . 
Lead  wir.3,    .... 
Mercury,      .... 

Platina, 

Silver, 

Steel,  s<.fi,    .... 

•'  razor-teaporYMi, 
Tin,  caet,  .... 
Zinc,  fused,  .     . 

"    roUed,      .    .     . 


S33 


Ash,  English,   .     . 


Birch,  English, 

"       Americf  n, . 
Box, 

Cedar,  Canadian,  . 

Chestnut.     .     .     . 

Deal,  Christiania  mi(f  i' », 
"     Memel  " 

"    Norwav  Spruce, 
"    English.  .... 

Elm.  seasoned,      .     .     . 

Fir,  New  England,     .     . 

"  Riga,  .  .  . 
Lignum-vitse,  .  . 
Mahogany,  Spanish, 


Specific 
Gravity. 


8.399 

8.6' )7 

S.S64  F. 

8.37S 
19.2.3>Ha 
19.361  Ua 

7.066  H 
7.046  H. 
7.29.5  H. 
7.2:id  H. 
7.079  H. 
6.99S  H. 

7.700 


7.473  F. 
7.740  F. 
7.S0.5  F. 
7.722  F. 


7.727  F. 


II  446 M. 
P.:J17 
]l3.5i>S  W. 
l.Ta'XilLv 
1^2.669  Ua 


l0  474H.i 

7.7S0 
7.840 

7  050  TV. 
'.'.540  W 


AV'eight 

per 

Cubic 

Foot 

in  lbs 


.760  B 

.792  B. 

.&iS  B. 
.960  B 

909  C 

.6-57  Ro. 
.69S  B. 
.590  B 
.340 
.470 
,553  B 
.553  B. 

.753  B. 

1.220 

.800 


Tensile 
I  Strength 
per  Square 
ilnchinlbs. 


524.94 
537.94 
554.00 
554.87 
1203.62 
1210.06 

441.62 
440.37 
455.94 
451, SI 
442.44 
437.37 

431.25 


467.37 
4S3.75 
4S7.S1 
432  62 


482.94 


715.37 
707.31 
S49.S7 
'218.75 
lli6.81 
65L62 
43G.25 
490.')0 
451  63 
1^0  JO 

47i  2.> 


17963  R. 
19072 
32S26  F. 
6122S 


16653  II. 
13505  H. 

21907  H. 
17466  H. 
13434  n. 

57120  L. 
61960  T. 
64960  T. 
5S134F. 
5S661  F. 
52099  F. 
47909  F. 

80214  T. 
S41S6  F. 
733-8  F. 
89162  F. 

1324  R. 

2531  M. 


40902  M. 
120000 
IrOOOO 
5322  M. 


47.53' 

4'^.50; 

4n.50' 
60.00' 

56.81' 

41.06! 
43.62 
36.87 
21.25 
29.37 
34.5G 
34.-56 

47.06 

76.25 

50.J0 


2GC<JG  ■» 

1I40CB^. 

1 3.300  Ro. 
12400 

17600 
7000 
13439 M. 

12000  B. 
IISOOM. 
16500 


Crushing 

Force  per 

Square 

Inch 

in  lbs. 


10637511. 
103540 U. 

14.54.3511. 
93335 II. 
86397  H. 

56000  ?G. 


fS633H.) 
\ 9363  H. ) 
f  3297 II. » 
J6102H.( 
11 663 II. 
S771  H. 


1033i B 


Moduliis 
of  Rup- 
ture S 
in  lbs. 


38556  H. 
37503  H. 
36288  H. 
43497  H. 
37503  H. 
35316  H. 

54000  G. 


12156  B 

10920  B. 
9624  B 


9364  B. 
10386  B. 


i  S^7S  B. 
'  e612B. 


{Il^,jl  «-"■ 


(8193 F. 
i  8193  H. 


I 


TABLE    VIII.       PROPERTIES    OF    MATERIALS. 


125 


Miiterials. 


Specific 
Gravitj'. 


Woods. 
Oak,  English,    .     . 

"     Canadian, 

Fine,  pitch, . 


red,     .  ... 

American,  white, 
"  Southern 


Poplar, 
ITeak,  . 


Other  Materials. 


Brick,  red,   .     . 
"      Dale  red. 


Chalk, 


Coal,  Penn.  anthracite, 

"         "      semi-bituminous, 
"     Md.  " 

Penn.  bituminous, 
Ohio  " 

"     English       " 
Earth, 

loamy  hard-stamped,  fresh, 

"  "  dry, 

garden,  fresh,    .     . 

"       dry,      .     . 

dry,  poor,      .     .     . 

Glass,  plate,      .     .     . 

Gravel, 

Gi-anite,  Aberdeen,    . 
Ivory,       


Limestone,  .... 

Marble,  white  Italian, 
black  Gal  way, 
Masoni-y,  quarry  stone,  dry, 
sandstone,  " 

"        brick,  dry,      .       | 

Ropes, 
hemp,  under  1  inch  diam., 
'*       from  1  to  3  in.     " 
"       over  3  inches      " 

Sand,  river, 

Sandstone, | 

"    Dundee,      .... 
"    Derby,  red  and  friable, 

Slate,  Welsh, 

"     Scotch, 


.931  B. 
.872  B. 
.6S0  B. 

.657  B. 

.455  Br. 
.372  Br. 

.333  M. 

.745  B. 


2. 1  GSR. 
2.0:^5  R. 
2.7,S4 
1.S69 
1..327Ta 
1.700  Ta 
1.453  Ta. 
1..552Ta. 
1.312  Ta. 
1.270  Ta. 
1.2.59  Ta. 

2.060  W. 
1.930  W. 
2.05 )  W. 
1.630  W. 
1.340  W. 
2.453 
1.920 
2.625  R. 
1.826 
2.400  W. 
2.S60  W. 
2.63S  H. 
2.695  H. 
2.400  W. 
2.050  W. 
1.470  W. 
1.590  W. 


1.8S6 
1.900  W. 
2.700  W. 
2.530  R. 
2.316  R. 
2.S83 


Weight 

per 

Cubic 

Foot 

iu  lbs 


58.3/ 


54.50 
41.25 

41.06 

2S.44 
54.50 

23.94 

46.56 


135..50 

130.31 

174.00 

116.81 

82.94 

106.25 

90.81 

97.00 

82.00 

79.3 

73.69 

!2>.Vo 
12  ).62 
128.12 
101.87 
83.75 
153.31 

1 20.no 

164.06 
114.12 
1.50.00 
173.75 
164.87 
168.44 
150.00 
128.12 
91.87 
99.37 


Tensile 

Strength 

per  Square 

Inch  in  lbs. 


117.87 
118.75 
168.75 
1.58.12 
144.75 
180.50 


10000  B. 
10253 
7318  M. 


7200  Be. 
15000  B. 


280 
300 


9420 
16626 


Crushing 
Force  per 

Square 
Inch 

in  lbs. 


(6184  II.) 

1 1005-n) 

14231  II.  i 
)  5982 II.  j 
6790  H. 1 
6790  II. ) 
(5395H.) 
\7518U.J 


(310711. 

I  5124  II. 

12101  II. 


SOSR. 
562  R. 

501  R. 


9230  W. 
7213  W. 
5156  W. 


10914  R. 

1500  W. 
6000  W. 
9583  G. 


Modulus 
of  Rup- 
ture S 
in  lbs. 


10032  B. 

10596  B. 

9792  B 

8046  B. 

7829  Br. 
1.39S7Br. 

14772  B. 


340  W. 

ISO  w. 


700  W. 
1700  W 
1062 
2664 


12800 
9600 


1400  W. 

600  W. 

13000  W. 

800  W. 

6630  R. 

3142  R. 

126  TABLE    IX.       MAGNETIC    VARIATION. 

TABLE    IX. 
MAGNETIC   VARIATION. 

The  following  table  has  been  made  up  from  varioi^  sources,  prin- 
cipally, however,  from  the  results  of  the  United  States  Coast  Survey, 
kindly  furnished  in  manuscript  by  the  Superintendent,  Prof  A.  D 
Bache.  '•  These  results,"  he  remarks  in  an  accompanying  note,  "  are 
from  preliminary  computations,  and  may  be  somewhat  changed  by  the 
final  ones."  Among  the  other  sources  may  be  mentioned  the  Smith- 
sonian Contributions  for  1852,  Trans.  Am.  Phil.  Soc.  for  1846,  Lond. 
Phil.  Trans,  for  1849,  Silliman's  Journal  for  1838,  1840,  1846,  and 
1852,  and  the  various  American,  British,  and  Russian  Government 
Observations.  The  latitudes  and  longitudes  here  given  are  not  always  • 
to  be  relied  on  as  minutely  correct.  Many  of  them,  for  places  in  the 
Western  States,  were  confessedly  taken  from  maps  and  other  uncer- 
tain sources.  Those  of  the  Coast  Survey  Stations,  however,  as  well 
as  those  of  American  and  foreign  Government  Observatories  and  Sta- 
tions, are  presumed  to  be  accurate. 

It  will  be  seen  that  the  variation  of  the  magnetic  needle  in  the 
United  States  is  in  some  places  west  and  in  others  east.  Tlie  line  of  no 
variation  begins  in  the  northwest  part  of  Lake  Huron,  and  runs  through 
the  middle  of  Lake  Erie,  the  southwest  corner  of  Pennsylvania,  the 
central  parts  of  Virginia,  and  through  North  Carolina  to  the  coast. 
All  places  on  the  east  of  this  line  have  the  variation  of  the  needle 
west,  —  all  places  on  the  west  of  this  line  have  the  variation  of  the 
needle  east ;  and.  as  a  general  rule,  the  farther  a  place  lies  from  this 
line,  the  greater  is  the  variation.  The  position  of  the  line  of  no  varia- 
tion given  above  is  the  position  assigned  to  it  by  Professor  Loomis  for 
the  year  1840.  But  this  line  has  for  many  years  been  moving  slowly 
westward,  and  this  motion  still  continues.  Hence  places  whose  varia- 
tion is  west  are  every  year  farther  and  farther  from  this  line,  so  th&t 
the  variation  west  is  constantly  increasing.  On  the  contrary,  places 
whose  variation  is  east  are  every  year  nearer  and  nearer  to  this  line, 
so  that  the  variation  east  is  constantly  decreasing.  The  rate  of  this 
increase  or  decrease,  as  the  case  may  be,  is  said  to  average  ab:3Ut  2'  for 
the  Southern  States,  4'  for  the  Middle  and  Western  States,  and  6'  for 
the  New  England  States.*  The  increase  in  "Washington  in  1840-2 
was  3'  44.2";  in  Toronto  in  1841-2  it  was  4'  46  2".    The  changes  in 

•  Prof  Loomis  in  Silliman's  Journal.  Vol.  XXXIX..  1R40. 


TABLK    IX.       MAGNETIC    VARIATION. 


127 


Cambridge,  1708, 
1742, 
1757, 
1761, 
1763, 
1780, 
1782, 
1783, 


u 
{( 

(( 


6  22 

7  30 

8  51 

9  18 


Cambridge,  Mass.  maybe  seen  from  tbe  following  determinations  of  the 
variation,  taken  from  the  Memoirs  of  the  American  Academy  for  1846. 

9     0  Cambridge,  1788,       6  38 

8     0  Boston,         1793,       6  30 

7  20  Salem,  1805,       5  57 

7   14  "  1808,       5  20 

7     0-  "  1810, 

7     2  Cambridge,  1810, 

6  46  "  1835, 

6  52  ''  1840, 

But  besides  this  change  in  the  variation,  which  may  be  called  secu- 
lar, there  is  an  annual  and  a  diurnal  change,  and  very  frequently  there 
are  irrc^-ular  chanires  of  considerable  amount.  With  respect  to  the 
annual  change,  the  variation  west  in  the  Northern  hemi.>pbere  is  gen- 
erally found  to  be  somewhat  greater,  and  the  variation  east  somewhat 
less,  in  the  summer  than  in  the  winter  months.  The  amount  of  this 
change  is  different  in  different  places,  but  it  is  ordinarily  too  small  to 
be  of  any  practical  importance.  The  diurnal  change  is  well  deter- 
mined. At  Washington  in  1840-2,  the  mean  diurnal  change  in  the 
variation  was,*  — 


Summer,  10  4.1 


Autumn,  6  21.2     Winter,  5  9.1      Spring,  8  10.7 


At  Toronto  the  means  were,  t  — 


t 

1841. 

6.67 

9.46 

12.38 

1843. 

1845. 

1847. 

1849. 

1850. 

1851. 

Winter, 

Spring  and  Autumn, 

Summer, 

5.64 

9.36 

11  70 

5.73 

9.15 

13.36 

7.28 
10.08 
13.84 

8.25 
12.25 
14.80 

8.01 
10.90 
13.74 

7.01 
10.82 
12.61 

The  diurnal  change  in  the  variation  is  such  that  the  north  end  of  the 
needle  in  the  Northern  hemisphere  attains  its  extreme  westerly  posi- 
tion about  2  o'clock,  P.  M.,  and  its  extreme  easterly  position  about 
8  o'clock,  A.  M.  In  places,  therefore,  whose  variation  is  west,  the 
maximum  variation  occurs  about  2  P.  M.,  while  in  places  whose  vari- 
ation is  east,  the  maximum  variation  occurs  about  8  A.  M.  In  Wash- 
ington, according  to  the  report  of  Lieutenant  Gilliss,  the  maximum  va- 
riation, taking  the  mean  of  two  years'  observations,  occurs  at  l*^-  33'"" 
P.  M.,  the  minimum  at  s'^-  6"-  A.  M. 

The  determinations  of  the  Coast  Survey  are  distinguished  by  the 
letters  C.  S.  attached  to  the  name  of  the  observer.  In  some  instances 
the  name  of  the  nearest  town  has  been  added  to  the  name  of  the  Coast 
Survey  station. 

*  Lieut.  Gilliss's  Report,  Senate  Document  172,  1845 
'■  London  Philosophical  Transactions.  1852 


1-^8 


TABLE    IX. 


MAGxXETlC    VARIATIUJN. 


Place. 


Maine. 
Agameuticus, 
Bethel, 

Bowdoin  Hill,  Port- 
land, 
Cape  Neddick,York 
Cape  Small, 
Kennebunkport, 
Kittery  Point, 
Mt.  Pleasant, 
Portland, 
Richmond  Island, 

Neiv  Hampshire. 

Fabyan's  Hotel, 
Hanover, 
Isle  of  Shoals, 
Patuccawa, 
Unkouoouuc, 

Vermont. 
Burlington, 

Ma.'i.'^ac/uisetts. 

Annis-squani, 
Baker's  Island, 

Blue  Hill,  Milton, 

Cambridge. 
Chapp.'iquidick.Ed- 

gartown, 
Coddonsimi,Mar- 

blehead, 

Copecut  Hill, 

Dorchester, 
Fort  Lee,  .Salem, 
Ilyaunis. 
Indian  Hill, 
Little  Xahant, 
Nantasket, 
Nantucket, 
New  Bedford, 
ShootHying    Hill, 

Barnstable, 
Tarpaulin  Cove, 

Rhode  Island. 

Beacon-pole  Hill, 

McSparran  Hill, 
Point  Judith, 

Spencer  Hill, 

Connecticut. 

Black  Rock,  Fair- 
field, 
Bridgeporc, 
Fort  Wooster, 
Groton  Point,  New 


London, 


Lati- 
tude. 

Longi- 
tude. 

Authority. 

Date 

A     ' 

o     * 

43  13.4 

70  41.2 

T.  J.  Lee,  C.  S. 

Sept.,  1817 

44  2S.0 

70  51.U 

J.  Locke, 

•lune,  1S45 

43  33.8 

70  16.2 

J.  E.  Ililgard.  C  S. 

Aug.,  1S51 

43  11.6 

70  36.1 

J.  E.  Hilgard;  C.  S. 

Aug.,  1851 

43  46.7 

69  50.4 

G.  \V.  Dean,  C.  S. 

Oct.,    1851 

43  21.4 

70  27.S 

J.  E.  Ililgard,  C.  S. 

Aug.,  1851 

43    4.S 

70  43.3 

J.  E.  Ililgard,  G.  S. 

Sept.,  1850 

44     1.6 

70  49.0 

G.  W.  Dean,  0.  S. 

Aug..  1 851 

43  41.0 

70  20.5 

J.  Locke, 

June,  1  "45 

43  32.4 

70  14.0 

J.  E.  Hilgard,  C.  S 

Sept.,  1S50 

44  16.0 

71  29.0 

J.  Locke, 

June,  1>45 

43  42.0 

72  10.0 

Prof  Young, 

1  S3  J 

42  59.2 

70  36.5 

T.  J.  Lee.  C.  S. 

Aug  ,  1847 

43    7.2 

71   11.5 

G.  W.  Dean,  C.  S. 

Aug.,  1849 

42  59.0 

71  35.0 

J.  S.  Ruth,  C.  S. 

Oct ,    1848 

44  27.0 

73  10.0 

J.  Locke, 

June,  1845 

42  .39.4 

70  40.3 

G.  W.  Keely,  C.  S. 

Aug.,  1849 

42  32.2 

70  46.8 

G.  W.  Keely,  C.  S. 

Sept.,  1S49 

42  12.7 

71     6.5 

T.  J.  Lee,  0.  S.         j 

Sept.  and  ) 
Oct.,  1845  J 

42  22.9 

71     7.2 

W.  C.  Bond, 

1352 

41  22.7 

70  23.7 

T.  J.  Lee,  C.  S. 

July,  1816 

42  31.0 

70  .50.9 

G.  W.  Keely,  C.  S. 

Sept,  1^49 

41  43.3 

71     3.3 

T.  J.  Lee,  C.  S.         { 

Sept  and  I 
Oct , 1844  1 

42  19.0 

71     4.0 

W.  C.  Bond, 

1839 

42  31.9 

70  52.1 

G.  W.  Keely,  C.  S. 

Aug.,  1849 

41  3S.0 

70  IS.O 

T.  J   Lee,  C.  S. 

Aug.,  1S46 

41  25.7 

70  40.3 

T.  J.  Lee,  C.  S. 

Aug.,  1S46 

42  26.2 

70  55.5 

G.  AV.  Keelv,  C.  S. 

Aug.,  1-^49 

42  18.2 

70  54.0 

T  J.  Lee,  C.  S. 

Sept.,  1847 

41   17.0 

70     6.0 

T  J.  Lee,  C.  S. 

July,  1346 

41  33.0 

70  54.0 

T.  J.  Lee,  C.  S. 

Oct.,    1845 

41  41.1 

70  20.5 

T.  J.  Lee.  C.  S. 

Aug.,  1 846 

4i  23.1 

70  45.1 

T.  J.  Lee,  C.  S. 

Aug.,  1846 

41  59.7 

71  26.7 

T.  J  Lee,  C.  S.        { 

Oct.    and  ) 

Nov.,  1844} 

41  29.7 

71  27.1 

T.  J.  Lee,  C.  S. 

July,  1844 

41  21.9 

7)   28.9 

R.H.Fauntleroy,C.S. 

Sept ,  1847 

41  40.7 

71  29.3 

T.  J.  Lee,  C.  S          { 

July  and  ) 
Aug.  1844  j 

41     S.6 

73  12.6 

J.  Renwick,  C.  S. 

Sept.,  1845 

4i   10.0 

73  11.0 

J.  Renwick,  C.  S. 

Sept.,  1845 

41   16.9 

72  53.2 

J.  S.  Ruth,  C.  S. 

Aug.,  1843 

41   18.0 

72    0.0 

J.  Renwick,  C.  S. 

Aug.,  1845 

Variation. 


o 
10 
11 

II 
11 
12 
11 
10 
14 
11 
12 


iO.OW. 
50.0  " 


41.1 
9.0 
5.5 
23.6 
30.2 
32.0 
28.3 
17.9 


11  32.0  W. 

9  15.0  " 
10  .3.4  " 
10  42.9  " 

9    5.6  " 


9  22.0  W. 


11  36.7  W. 

12  17.0  " 

9  13.8  « 
10    8.0  " 

8  47.7  » 


49.8 
12.1 


2.0 

14.5 

22.0 

49.3 

40.9 
9  33.5 
9  14.0 
8  54.6  " 


9  40.1 
9  10.1 


9  29.8  W. 

8  53.3  " 

8  59.4  " 

9  11.9  » 


6  53.5  W. 

6  19.3  " 

7  26.4  " 

7  29.5  " 


TABLE    IX. 


MAGNETIC    VARIATION. 


129 


Place. 

Lati-w^ 
tude. 

Longi- 
tude. 

Authority. 

Date. 

4 

Variation 

O        ( 

O        ( 

o 

1 

Milfovd, 

41    IG.O 

73     1.0 

J.  Renwick,  C  S. 

Sept  ,  1S45 

6 

.3'-.3  W 

New  llaveu,  Pavil- 

ion, 

41    18.5 

72  55.4 

J.  S.  Ruth,  C.  S 

Aug.,  1848 

6 

37.5  " 

New   Haven,    Yale 

College, 

41   1S.5 

72  55.4 

J.  Renwick,  C.  S. 

Sept.,  1845 

6 

17.3  " 

Nojwalk, 

41     71 

73  24.2 

J.  Renwick,  C.  S. 

Sept.,  1S44 

6 

46.3  " 

1  Oyster  Point,  New 

i     Haven, 

41    17.0 

72  55.4 

J.  S.  Ruth,  0.  S. 

Aug.,  1843 

6 

32.3  " 

'■jachenrs    Head, 

Guilford, 

41   17.0 

72  43.0 

J.  Renwick,  C.  S. 

Aug.,  1S45 

6 

15.2  " 

Sawpits, 

40  59  5 

73  o9.4 

J.  Renwick,  C.  S. 

Sept.,  1344 

6 

1.6  " 

Say brook. 

41    16.0 

72  20.0 

.1.  Renwick,  C.  S. 

Aug.,  1845 

6 

49.9  " 

Stamford, 

41     3.5 

73  32  0 

J.  Renwick,  C.  S. 

Sept.,  1844 

8 

40.4  " 

Stouiugton, 

41  20.0 

71  54.0 

J.  Renwick.  C.  S. 

Aug.,  1845 

7 

3^.2  " 

Netv  York. 

Albany, 

42  39.0 

73  44.0 

Regents'  Report, 

1836 

6 

47.0  W. 

lllooiuingdale  Asy- 

hnii, 

40  43.8 

73  57,4 

J.  Locke,  C.  S. 

April,  1846 

5 

10  9  " 

Cole,  Staten  Island, 

10  31.8 

74   13.^ 

J.  Locke,  0.  S. 

April,  IS46 

5 

33.8  " 

!  Drowned    Meadow. 

i      L.  I., 

40  .56.1 

73     3.5 

J.  Renwick,  C.  S. 

Sept.,  1845 

6 

3.6  " 

Flatbush,  L.  L, 

40  40  2 

73  57.7 

J.  Locke,  C.  S. 

April,  1 846 

5 

54.6  " 

Greenport,  L.  1., 

41     6.0 

72  21.0 

J.  Jlenwick,  C.  S. 

Aug.,  IS45 

7 

14.6  " 

Leggett, 

40  4^9 

73  53  0 

R.H.  Fauntleroy,C.S. 

Oct.,    1847 

5 

40.6  " 

Lloyd's     Harbor, 

L.  I., 

40  55.6 

73  24. S 

J.  Renwick,  C.  S 

Sept.,  1844 

6 

12.5  " 

New  lloehelle, 

40  52.5 

73  47.0 

J.  Renwick,  C.  S. 

Sept.,  1844 

5 

31.5  " 

New  York, 

40  42.7 

74     0  ! 

J.  Renwick;  C.  S. 

Sept.,  1845 

6 

25.  n  » 

Oyster  Bay,  L.  I., 

40  52.3 

73  31  3 

J.  Renwick,  C.  S. 

Sept.,  1344 

6 

53  G  " 

L'ou.^e's  Point, 
Sand.s   Lighthouse, 

45     0.(1 

73  21.0 

Boundary  Survey, 

Oct.,    1845 

11 

2S.0  " 

i 

L.  I., 

40  51.9 

73  4.3.5 

R.H.  F:uintlcrov,C.S. 

Oct.,    1847 

6 

9.7  "     1 

Sands  Point,  L.  I., 

40  .52.0 

73  43.0 

J.  Renwick,  C.  "S. 

Sept.,  1845 

7 

14.6  "     i 

\^'atchhill.  Fire  Isl- 

li 

and, 

40  41.4 

72  53  9 

R.H.  Fauntlcroy,C.S. 

Oct.,    1847 

7 

33  5  <^    ii 

West  Point. 

41  25.(1 

73  56  0 

Prof.  Davies, 

Sept.,  1835 

6  32.0  " 

Neiv  Tcrstij. 

Oape    5Iay    Light- 

'     house. 

38  55  8 

74  57.6 

J.  Locke,  C.  S. 

June,  1346 

3 

3.2  AY. 

('Iiew, 

39  43.2 

75     9  7 

J.  Locke,  0.  S. 

July,   1316 

3 

20.4  " 

Oiiurch  Landing, 

39  40  9 

75  30.3 

J.  Locke,  0.  S. 

June,  1346 

*5 

45.8  « 

Egg  Island, 

39  10.4 

75     7.8 

J.  Locke,  0.  S. 

June,  1346 

3 

13.2  " 

Hawkins, 

.39  25.5 

75   17.1 

J.  Locke,  C.  S. 

June,  1346 

2 

.53.7  " 

Mt.Piosc,  Princeton, 

40  22.2 

74  42.9 

J.  E.  Hilgard,  C.  S. 

Aug.,  1852 

5 

31.8  » 

Newark, 

40  44. '^ 

74     7.1) 

■T.  Locke,  C.  S- 

April,  1346 

5 

32.7  " 

Pine  Mountain, 

39  25.0 

75   19  9 

J.  Locke,  C.  S. 

June,  1346 

2 

52.0  » 

Port  Norris. 

39  14.5 

75     1.0 

J.  Locke.  (".  S. 

June,  1346 

.3 

6.5  « 

Sandy  llDok, 

40  28.0 

73  59.  S 

J.  Renwick,  C.  S 

Aug.,  1344 

5 

54  0  " 

Town   Bank,   Cape 

May, 

39  .58.6 

74  57.4 

.7.  Locke.  C.  S. 

June,  1846 

3 

3  2  " 

Tucker's  Island, 

39  30.  S 

74  16.9 

T.  J.  Lee,  0.  S. 

Nov.,  1846 

4 

23.8  " 

White     Hill,    Bor- 

■*    ' 

dentown, 

40    8.3 

74  43  8 

J.  Locke,  C  S. 

April,  1846 

4  22.5  " 

Pennsylvania. 

Girard     College, 

Philadelphia, 

39  58.4 

75     9.9 

J.  Locke,  C.  S. 

May,    1346 

3 

50.7  W. 

Pittsburg, 

40  26.0 

79  .53.0 

J.  Locke, 

May,    ls45 

0 

33.1  " 

Vauuxeni,  Bristol,  |40     5.9 

74  52.7 

J.  Locke,  C.  S. 

July,  1346 

4 

20.5  "     1 

*  Loeal  ittrictinn  exi.5t.=?  here,  according  to  Prof.  Locke. 
7 


130 


TABLE    IX.       MAGNETIC    VAEIATION. 


Place. 

Lati- 
tude. 

Longi- 
tude. 

Authority.    "^ 

Date. 

Variation. 

Delaivare. 

Bombay      Hook 

o     / 

O        1 

o 

Lighthouse, 

39  21.8 

75  30.3 

J.  Locke,  C.  S 

June,  1846 

3  17.9  W 

Fort^Delaware,  Del- 

aware River, 

39  35.3 

75  33.8 

J.  Locke,  C.  S. 

June,  1846 

3  16.0  " 

Lewes  Lauding, 

3S  48.8 

75  11.5 

J.  Locke,  C.  S. 

July,   1846 

2  47.7  " 

Pilot  Town, 

.33  47.1 

75     9.2 

J.  Locke,  C.  S. 

July,  1346 

2  42.2  » 

Sawyer, 

.39  42.0 

75  .3.3.5 

J.  Locke,  C.  S. 

June,  1346 

2  47.8  " 

Wilmingtv.n, 

.39  44.9 

75  33.6 

J.  Locke,  C.  S. 

May,   1S46 

2  31.8  « 

Manjlnnd. 

Annapolis, 

33  56.0 

76  35.0 

T.  J.  Lee,  C.  S. 

June,  1845 

2  14.0  W. 

Bodkiu, 

39    8.0 

76  25.2 

T.  J.  Lee,  C.  S. 

April,  1817 

2    2.6  « 

Finlay, 

39  24.4 

76  31.2 

J.  Locke,  C.  S. 

AprU, 1846 

2  19.5  " 

Fort     McIIenry, 

Baltimore, 

39  1.5.7 

76  .34.5 

T.  J   Lee,  C.  S. 

April,  1347 

2  13.0  » 

Hill, 

35  53.9 

76  52.5 

G.  W.  Deau,  C.  S. 

Sept.,  1850 

2  1.5.4  " 

Kent  Island, 

39     1.8 

76  18.8 

J.  Ileustou.  C.  S. 

July,  1349 

2  30.5  " 

Marriott's, 

33  52.4 

76  36.3 

T  J    Lee,  C.  S.       - 

June,  1549 

2     5.2  " 

North  Point, 

•39  11.7 

76  26.3 

T   J.  Lee.  C.  S. 

July,  1846 

1  42.1  " 

Osborne's  Ruin, 

39  27.9 

76  16.6 

T  J.  Lee,  C.  S. 

June,  1845 

2  32.4  « 

Poole's  Island, 

39  17.1 

76  15.5 

T  J.  Lee,  C.  S. 

June,- 1847 

2  23.5  « 

llosaune. 

39  17.5 

76  42.8 

T.  J.  Lee.  C.  S. 

June,  1815 

2  12,0  " 

Soper, 

39     5.1 

76  56.7 

G.  W.  Deau,  C.  S. 

July,  1350 

2     7.0  " 

South    Base,   Kent 

Islaud, 

33  53.S 

76  21.7 

T.  J.  Lee,  C.  S. 

June,  1845 

2  26.2  " 

SusquehannaLight- 

house,  Havre  de 

Grace, 

39  32.4 

76    4.8 

T  J.  Lee,  C.  S. 

July,  1817 

2  51.1  « 

Tavlor, 

33  59.  S 

76  27.6 

T  J.  Lee,  C.  S. 

May,   1347 

2  18.4  " 

Webb, 

39    5.4 

76  40.2 

G  W.  Dean,  C.  S. 

Nov.,  1350 

2    7.9  ' 

District  of  Colmn- 
bia. 

Oausten,      George- 

town, 

33  5.5.5 

77     4.1 

G.  W.  Dean,  C.  S. 

June,  1351 

2  11.3  W. 

Washington, 

33  53.7 

77    2.8 

J.  M.  Gilliss, 

June,  1342 

1  26.0  « 

Virginia. 

Charlottesville, 

33    2.0 

73  31.0 

Prof.  Patterson, 

1835 

0     0.0 

Roslyn,      Peters- 

burg, 

37  14.4 

77  23.5 

Q.  "W.  Dean,  C.  S. 

Aug.,  1852 

0  26.4  w^ 

Wheeling, 

40    8.0 

80  47.0 

J.  Locke, 

April,  1345 

2    4.0  E. 

North  Carolina. 

Bodie's  Island, 

35  47.5 

75  31.6 

C.  0.  Boutelle,  C.  S. 

Dec.,   1846 

1   1.3.4  W. 

Shellbank, 

3.     .3.3 

75  44.1 

C.  0.  Boutelle,  C.  S. 

Mar.,  1847 

1  44.8  " 

Stevenson's  Point, 

.36     6.3 

76  11.0 

C  0.  Boutelle,  G.  S. 

Feb.,  1847 

1  39.7  " 

South  Carolina. 

Breach  Inlet, 

.32  46.3 

79  48.7 

C.  0.  Boutelle.  C.  S. 

April,  1849 

2  16.5  E. 

Charleston, 

32  41.0 

79  53.0 

Capt.  Bamett' 

May,    1341 

2  24.0  " 

Ri.st  Base,  Edisto, 

.32  33.3 

80  10.0 

G.  Davidson,  C.  S. 

April,  1350 

2  53.6  " 

Georgia. 

Atliens, 

.34    0.0 

33  20.0 

Prof.  McCay, 

18.37 

4  31.0  E. 

Cohuubus, 

.32  2S.0 

85  10.0 

Geol.  Survey, 

1839 

5  30.0  " 

Milledgeville, 

.33    7.0 

83  20.0 

Geol.  Survey, 

1833 

5  51.0  " 

Savannah, 

1 

32    5.0 

31     5.2 

J.  E.  IlJlgard,  ?■.  S. 

April,  1852 

3  4.5.0  " 

TABLE    IX.       MA 

GNETIC    VARIAT 

ION. 

m\ 

r 

Place. 

Lati- 
tude. 

Longi- 
tude. 

Authority. 

Date. 

Variation. 

Florida. 

O      1 

4  25.2  E. 

5  20.5  " 
5  29.2  « 
5  29.0  » 

Cape  Florida, 
Cedar  Keys, 
St.  Marks  Light, 
Saud  Key, 

o     / 
25  39.9 
29     7.5 
iO     4.5 
21  27.2 

SO    9.4 
S3    2.8 
84  12.5 
81  52.0 

J.  E.  Ililgard,  C   S. 
J.  E.  Hilgard,  C.  S. 
J.  E.  Hilgard,  C.  S. 
J.  E.  Hilgard,  C.  S. 

Feb.,  1850 
Mar.,  1852 
April,  1852 
Aug.,  1849 

Alabama. 

Fort  IMorgan,  Mo- 
bile Bay, 
Tuscaloosa, 

30  13.8 
33  12.0 

SS    0.4 
87  42.0 

R.H.Fauntleroy,C.S. 
Prof.  Barnard, 

May,   ]3!7 
1839 

7    3.8  E. 
7  28.0  " 

Mississippi. 

East  Pascagoula, 

30  20.7 

88  31.4 

R.II.  Fauntleroy,C.S. 

June,  1847 

7  12.4  E. 

Texas. 

j 

Dollar  roint,   Gal- 
veston, 
Mouth  of  Sabme, 

29  2G.0 
29  43.9 

94  53.0 
93  5L5 

R.II.  Fauntleroy,C.S. 
J.  D.  Graham, 

April,  1848 
Feb.,   1840 

8  57.2  E. 
8  40.2  " 

Ohio. 

Carrolton. 

Cincinnati, 

Columbus, 

Hudson, 

Mai-ietta, 

Oxford, 

St.  Mary's, 

39  33.0 
39     6.0 

39  57.0 
41   15.0 
.39  26.0 
.39  .30.0 

40  32.0 

84     9.0 
84  22.0 

83  3.0 
81  26.0 

81  29.0 

84  33.0 
si  ly.c 

J  Locke, 
J.  Locke, 
J.  Locke, 
E.  Loomis, 
J.  Locke, 
J.  Locke, 
J.  Locke, 

Sept.,  1845 
April,  1845 
July,  1845 
1S4M 
April,  184.5 
Aug.,  1845 
Sept.,  1345 

4  45.4  E. 
4    4.0  " 
2  29.3  " 
0  52.0  " 

2  25.0  " 
4  50.0  " 

3  4.0  " 

Tennessee. 

\ 

Nashville, 

36  10.0 

86  49.( 

Prof.  Hamilton, 

1835 

7    7.0  B. 

Michigan. 

, 

Detroit, 

42  24.0 

82  58.0 

Geol.  Report, 

1840 

2    0.0  E. 

Indiana. 

Richmond, 
South  Hanover, 

39  49.0 
33  45.0 

&4  47.0 
85  23.0 

J  Locke, 
Prof.  Dunn, 

Sept.,  1845 
1837 

4  52.0  E 
4  35.0  " 

1 

Illinois. 

Alton, 

38  52.0 

90  12.0 

H.  Loomis, 

1840 

7  45.0  E. 

Missouri. 

. 

St.  Louis, 

33  36.0 

89  36.0 

Col.  NicoUs, 

1835 

8  49.0  E. 

Wisconsin. 

^ 

Madison, 
Prairie  du  Chien, 

43    5.0 
43     1.0 

89  41.0 
91     8.0 

U.  S.  Surveyors, 
U.  S.  Surveyors, 

Nov.,  1839 
Oct.,   1839 

7  30.0  E. 
9    5.0  " 

loioa. 

Brown's  Settlement 
Davenport, 
Farmer's  Creek, 

42    2.f 

41  30.C 

42  13.C 

91  J8.0 

90  34.0 

1    90  39.  C 

J.  Locke, 

U.  S.  Surveyors, 

J.  Locke, 

Sept.,  1839 
Sept.,  1839 
Oct.,   1839 

9     4.0  E. 
7  50.0  " 
9  11.0  " 

1 

Wapsipinnicon 
River, 

41  44.C 

1    90  39.C 

J.  Locke, 

Sept.,  1839 

8  25.0  « 

Cnlifornia. 

Point  Conception, 

34  26.C 

I  120  26.f 

)  G.  Davidson,  C.  S. 

Sept.,  1850 

113  49.5  E. 

b. 

. 

15!^ 

TABLE 

IX.       MAGNETIC    VARIATION. 

Place. 

Lati- 
tude. 

Longi- 
tude. 

Authority. 

Date. 

Variation. 

Point     Pinos, 

O       1 

o      / 

o 

/ 

Monterey, 

36  33.0 

121  54.0 

G.  Davidson,  C.  S. 

Feb.,  1351 

14 

53.0  E. 

PresiLlio,     San 

Francisco, 

37  47.8 

122  27.0 

G.  Davidson.  C.  S. 

Feb.,   IS52 

15 

26.9  " 

San  Diego, 

32  42.0 

117  14.0 

G.  Davidson,  C.  S. 

May,    1351 

12  29.0  « 

Oregon. 

Cape     Disap- 

pointment, 

46  16.6 

124    2.0 

0.  Davidson,  G.  S. 

July,  1351 

20 

45.0  E. 

Ewing  Harbor, 

42  44.4 

124  21.0 

G.  Davidson,  C  S. 

Nov.,  1351 

13 

29.2  «' 

Washington 

Territory. 

Scarboro'    Har- 

bor, 

I 

43  21.3 

124  37.2 

G.  Davidson,  C.S. 

Aug.,  1852 

21 

.30.2  E. 

BRiTisa    Amer- 

ica. 

Montreal, 

45  30.0 

73  35.0 

Capt.  Lefroy, 

1342 

8 

.53.0  W. 

Quebec, 

46  49.0 

71   16.0 

Capt.  Lefroy, 

1342 

14 

12.0  " 

St.  Johns,  C.  E. 

45  19.0 

73  13.0 

Capt.  Lefroy, 

1842 

11 

22.0  " 

StansteaJ, 

45    0.0 

72  1.3.  Q 

Boundary  Survey, 

Nov.,  1345 

11 

33.0  *' 

Toronto, 

43  39.6 

79  21.5 

British  Govern., 

Sept.,  1344 

1 

27.2  " 

New  Grenada 

Panama, 

8  57.2 

79  29.4 

\V  H.  Emory, 

Mar.,  1349 

6 

54.6  S. 

Eastern  Hemi- 

sphere. 

Green\vich,Eng- 

land. 

51  23.0 

0    0.0 

Prof.  Airy, 

1841 

23 

16.0  W. 

Makei-stoun, 

Scotland, 

55  35.0 

2  31.0  \Y. 

J.  A.  Broun, 

1342 

25 

2=!.0  « 

Paris,  France, 

43  50.0 

2  20.0  E.  1 

Paris  Observatory 

Nov.,  1851 

20 

25.0  « 

Munich,    Bara- 

ria, 

43     9.0 

11  .37.0  " 

1842 

16 

43.0  " 

St.    Peter.^burg, 

1 

1 

Russia. 

59  56.0 

30  19.0  « 

Russian  Govern., 

1842 

6  21.1  "   II 

Catherineuburg 

Siberia. 

■56  51.0 

60  ai.O  "   ; 

Russian  Govern., 

1842 

6 

33.9  B 

Xertchiusk,   Si- 

beria. 

51  56.0 

116  31.0  " 

Russian  Govern., 

1342 

3 

46.9  W. 

St.  Helena, 

15  .56.7  S. 

5  40.5  W.I 

British  Govern., 

Dec.,   1845 

23 

36.6  " 

Cape    of    Good 

Hope, 

33  56.0  '■' 

18  23.7  E. 

British  Govern  , 

.July,  1346 

29 

8.0  « 

Hobarton,    Van 

1 

Diemen-s  Ld., 

42  .52.5  • 

147  27.5  "  : 

British  Govern., 

Dec.,  1343 

10 

8.01. 

TABLE    X.       TRIGONOMETRICAL    FORMULA. 


133 


TABLE    X. 

TRIGONOMETRICAL  AND  MISCELLANEOUS  FORMULA 

Let  a  (fig.  57)  be  any  acute  angle,  and  let  a  perpendicular  B  Che 
irawn  from  any  point  in  one  side  to  the  other  side.     Tlien,  if  the  sidea 


Fig.  57. 


>f  the  right  triangle  thus  formed  are  denoted  by  letters,  as  in  the  fig 
arc,  we  shall  have  these  six  formula :  — 


1.     sin,  A  = 


2.     COS.  A  =  -  . 


3.     tan.  A  = 


4. 

cosec. 

-^-l 

5. 

sec. 

■^-l 

6. 

cot. 

a 

Given- 
a.  c 

a,  b 

J.,  a 

A,b 


10 

11  :.4.  c 


Solution  ofRi'jht  Triangles  (fig.  57). 
Sought. 
A,B,l 


A,  B,  c 
B,b,c 

B,  a,  c 
B,a,b 


Formulae. 

a 


sin.^=-,   cos.  C  =  -,    b=-^ic-\-a){c  —  a) 

c  c 


tan.  A  =  :^  ,        cot.  B  =  -^  ,    c  =  ya*  -f  b*. 


B  =.90°  —  A,     b  =  a  cot.  A,    c  =^ 
B  =  90o  —  A.     a  =  6tan. -1.     c  = 


sin.  A 
b 


COS.  A  ' 
B=--90°  —  A.      a  =  csin..4,      i  =  c  cos.  .4 


134 


TABLE    X.       TRIGONOMETRICAL    AND 


Solution  of  Oblique  Triangles  (fig.  58). 


Fig.  58 


12 
13 
14 

15 


16 
17 

18 


Given. 
A,  B,  a 

A,  a,  b 

a,b,  C 

a,  6,  c 


A,B,C,a 
A,  b,  c 

a,  b,  c 


Sought.  I 
b 

B 


b  = 


sin.  B 


a  sin.  B 
sin.  A    ' 

b  sin.  A 
a 


Formulae. 


A  —  Btan.^  {A  —  B) 


ja  —  b)  can,  i  (A  +  B) 


•Ifs=i(a  +  6  +  c),    sm.  ^A=^l^^'l^. 
cos4^=  J^\    tan.i^=  J(f^i^>, 

•^  y/         be  ^  >       5(5  — O) 


.  sin.  A  = 


2  .^A*  (5  —  a){s  —  b)  (s  —  c) 


be 


a-  sin.  B  sit    C 


area 
area 
area 


area  = 


2  sin.  A 
area  =  hbc  sin.  A. 


s=i  (a  4-  6  +  r,,    area=ys  {s—a)  {s—b)  {«-  «). 


General  Trigoriometri<v*'  Fomnilce. 


19! 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
•30 
!3I 


sin.' 


J.  +  cos.'^  yl  =  1. 


.=   9 


sin.  (J.  ±  B)  =  .sin.  ^1  cos.  75  ±  sir   h  cos.  J.. 
COS.  {A  da  B)  =  COS.  ^  COS.  B  rp  sin.  .^  "in.  B. 
sin.  2  A  =  2  sin.  ^  cos.  ^. 
COS.  2  J.  =  C0S.2  A  —  sin,2  J^  ==  i  _  o  siii      i 

sin."  A  =  h  —  k  ^os-  2  ^• 
COS.-  ^  ^  i  +  ^  COS.  2  ^. 

sin.  J.   +  sin.  B  =  2  sin.  ^  (^  +  S)  cos.  ^  (^        ZJ). 
sin.  .-1   —  sin.  B  =  2  cos.  ^  (^  +  5)  sin.^  [A       B). 
COS.  ^   +  cos.  B  =-2  cos.  |(A  +  JB)  cos.  ^  (J.  •     fJ). 
cos.  B  —  cos.  ^  =  2  sin.  ^  [A -\- B)  sin.  ^  (^  —  P) 
sin .2  A  —  sin.2jB=  cos.=  J5 -cos.^^  =  sin.  (^  +  ^)  sir   i 
COS.-  /I  —  sin.'-'  B  =  COS.  (A  +  S)  cos.  (4  —  B). 


COS.*  ^  —  I- 


MISCELLANEOUS    FORMULJE. 


195 


,       I  Bin.  A 

132  tan.  A  =  ^^;—^ 

COS.  A 
sin.  A 


33 


cot.  A 


tan  4  ±jan   B 


tan  ^  ±jan   /J 
34  tan.  (^  ±  i)  =  1  q:  t^.  J^tan.  B 

I  sin   (A  ±B) 

35|tan.  A  ±  tan.  B  =  ^^^  ^  cos.  B  ' 


36  cot.  A  ±  cot.  /i 
jsin  ^  +  sin    B 


38 


sin.  A  —  sin.  B 
sin  A  4-  sin-  >S 
COS.  A  -\-  COS.  .B 


sinj^±^) 
-^  sin   j4  tin.  B 
tan   ^{A^-  B) 
tan.  i  (4  —  B)  ■ 

tan.  H^  +  ^) 


sin  A  -\-  sin   J5  ^    \   i  a  n\ 

391 _,-* T  =  cot.  ^  (A  —  -U). 

icos    B  —  COS.  A  -  ^ 

sin.  -4.  —  sin.  B        ^  „    1  /  /i  R\ 

40 rn u  =  tan.  f  ( A  —  /j  • 

;co.-^   .4  +  cos  B  -^  ^ 

'sin.  A  —  sin.  B 

I  cos   B  —  COS.  A 

sin  A 

42  tan.  ^  A  =  1  +  cos.  i 


cot.  ^(^  +  1^1- 


43 


cot.  h  A  =  ^ 


sin.  .4 
—  cos   A 


Miscellaneous  Formidai. 

Sought.              1 

Given. 

Formom. 

Area  of 

44 

Circle 

Radius  =  r 

71  r^. 

45 

Ellipse 

Semi-axes  ==  a  and  b 

nab. 

46 

Parabola 

Chord  =  c,  height  =  h 

%ch* 

47 

Regular  Polygon 
Surface  of 

(  Side  =  a,  number  of  ) 
1  sides  =  «                     ) 

180° 
\  or  n  cot.    ^  • 

48 

Sphere 

Radius  =  r 

4  n  r"'. 

,'9 

Zone 

Radius  =  r,  height  =^  h 

2  71  r  h. 

M^adiusof  sphere=r  ) 

S— (Ji  -  2)180'- 

50 

Spherical  Polygon 
Solidity  of 

)  sum  of  angles  =  ^^  ( 
(  number  of  sides  =  n) 

■;i/''X         180  D 

.51 

Prism  or  Cylinder 

Base  =  b,  height  =  k 

bk. 

52 

Pyramid  or  Cone 

Base  =  b,  height  =  h 

^bh. 

53 

Frustum  of  Pyr-  ) 
amid  or  Cone  ) 

(  Bases  =  b   and   ftj ,  ) 
1  height  =  h                 ) 

kh{b-{-b,  +  ybb,) 

*  The  area  of  a  circular  segment  on  railroa^l  curves,  where  the  chord  is  very  long 
m  proportion  to  the  height,  may  be  found  with  great  accuracy  by  the  above  formula 


f36 


TA.BLE    X.       JIISCELLANEOUS    FOUMULiE. 


54 
55 


Sough., 
Solidity  of 

Sphere 


Given. 


Radius 


c;  1     •     le  J  i  TJi^dii   of  bases  =  r  ) 

'■  "I  and  /-,  ,  height  =  fi    ) 

-^    T5    1  ^    o   1        -1      f  Semi-transverse  axis " 
o6     Prohite  Spiicroul  ,.  ,,. 

'  ■  J      or  ellipse  =  a 

I  Semi  conjugate  ax 


Formulae. 


4  -i 

3  T  r\ 


58 


Oblate  Spheroid 


Paraboloid 


ixis 


[      of  ellipse 

j  Kadi  us  of  base  =  ?•,  I 
1  heiixht  ^  /i  ( 


( 


3  71  a^  b. 


*  ;r  r^  h. 


TT.  =  .3.U159  265.35  89793  23846  26433  83280. 
Log.  71  =  0.49714  98726  94133  85435   12682  88291 

United  States  Standard  Gallon  =  231         cnb.  in.    =    0.133681  cub. ft 


"  "  "         Bushel  =  21.50.42       " 

British  Imperial  Gallon  =  277.27384  " 

According  to  Ilassler. 
French  Metre,  =  3.2817431  ft., 

Litre,  =  61.0741569  cub.  in., 

Kilogram,        =  2.204737  lb.  avoir.. 

Weight  of  Cubic  Foot  of  Water, 

Barora.  30  inches.  Therm.  Falir.  39.83°, 


(C 


=     1.244456       " 
=     0.160459       " 

As  usually  given. 
=     3.280899  ft. 
=    61.02705  cub.  in. 
=    2.204597  lb.  avoir 

=     62.379  lb.  avoir. 
=     62.321        " 


Length  of  Seconds  Pendulum  at  Xcw  York        =    39.10120  inches. 
''        "         "  "  "  London  =    39.13908      " 

"  Paris  =    39.12843      " 

Equatorial  Radius  of  Earth  according  to  Bessel  =  20,923.597.017  feet 
Polar  "  •'  «  '■         =  20,853,654.177     ^ 


TABLE    XI. 


SQUARES,  CUBES,  SQUARE  ROOTS,  CUBE  ROOTS, 


AND 


RECIPROCALS   OF  NUMBERS 


T&OM  1  TO  1054. 


138    TABLE  XI.   SqUAKES,  CUBES,  SQUARE  ROOTS, 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.41421.36 

1.2599210 

.500000000 

3 

9 

27 

1.73205:J8 

1.44224<.6 

.3333.333.33 

4 

16 

64 

2.0000000 

1.. 53740 11 

.250000000 

5 

25 

125 

2.2360680 

1.7099759 

.200000000 

6 

36 

216 

2.4494897 

1.3171206 

.166666667 

7 

49 

343 

2.6457513 

1.9129312 

.142857148 

8 

64 

512 

2.82S4271 

2.0000000 

.12.5000000 

9 

81 

729 

3.0000000 

2.0800337 

.111111111 

10 

100 

1000 

3  1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166243 

2.2239801 

.090909091 

12 

144 

1723 

3.4641016 

2.2894286 

.033.333333 

13 

169 

2197 

3.605.5513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.071428571 

15 

225 

3375 

3.8729833 

2.4662121 

.066666667 

16 

256 

4096 

4.0000003 

2.5198421 

.062500000 

17 

239 

4913 

4.1231056 

2..57128I6 

.053823529 

13 

324 

5332 

4.2426107 

2.6207414 

.0555.55556 

19 

361 

6859 

4.3588989 

2.6634016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

9261 

4.5325757 

2.7589243 

.047619048 

22 

434 

10643 

4.6904153 

2.3020393 

.045454545 

23 

529 

1216/- 

4.7953315 

2.8433670 

.04347326 

24 

576 

13324 

4.3939795 

2.8344091 

.041666667 

25 

625 

15625 

5.0000000 

2.9240177 

.010000000 

26 

676 

17576 

5.0990195 

2.9624960 

.033461533 

27 

729 

19633 

5.1961524 

3.0000000 

.037037037 

23 

784 

21952 

5.2915026 

3.0365339 

.035714236 

29 

841 

5^4339 

5.3851643 

3.0723163 

.034482759 

30 

900 

27000 

5.4772256 

3.1072.325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.0.32253065 

32 

1024 

32763 

5.6563542 

3.1743021 

.031250000 

33 

1039 

35937 

5.7445626 

3.2075313 

.030303030 

34 

1156 

39304 

5.S309519 

3.2396113 

.029411765 

35 

1225 

42875 

5.9160793 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

506.53 

6.0527625 

3.-33222 13 

.027027027 

33 

1444 

54372 

6.1644140 

3.3619754 

.026315739 

39 

1521 

59319 

6.2449930 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1631 

63921 

6.4031242 

3.4432172 

.024390244 

42 

1764 

74033 

6.4307407 

3.4760266 

.023809524 

43 

1349 

79507 

6.5574335 

3.503.3931 

.0232.55314 

44 

1936 

85134 

6.6332496 

3.5303483 

.022727273 

45 

2025 

91125 

6.7032039 

3.5568933 

.022222222 

46 

2116 

97336 

6.7323300 

3.583)479 

.021739130 

47 

2209 

103823 

6.3556.546 

3.6038261 

.021276600 

43 

2304 

110592 

6.9232032 

3.6:342411 

.020333333 

49 

^01 

117649 

7.0000000 

3.6593057 

.020403163 

50 

2500 

125000 

7.0710673 

3.6340314 

.020000000 

51 

2601 

132651 

7.141-4234 

3.7084298 

.019607843 

52 

2704 

140603 

7.2  LI  1026 

3.7.325111 

.019230769 

53 

2309 

143377 

7.2801099 

3.7562858 

.018367925 

51 

2916 

157464 

7.3484692 

3.7797631 

.013518519 

55 

3J25 

166375 

7.4)61935 

3.3029525 

.013131818 

56 

3136 

175616 

7.4833143 

3.3258624 

.017357143 

57 

3249 

185193 

7.5493344 

3.8485011 

.017543360 

53 

3364 

195112 

7.0157731 

3.8703766 

.017241379 

59 

3481 

205379 

7.6311457 

3.8929965 

.016949153 

60 

3600 

216000 

7.7459667 

3.9143676 

,016666667 

61 

3721 

226931 

7.8102497 

3.9364972 

.016393443 

62 

3344 

233323 

7.3740079 

3.9573915 

.016129032 

CUBE    ROOTS,    AND    HECirilOCALS. 


139 


lU 


63 

64 
65 
66 
67 
6S 
6^ 

70 
71 
72 
73 

74 
7o 
76 

77 
73 
79 

30 
81 
82 
83 
84 
85 
86 
87 
88 
89 

90 
91 
92 

93 
94 
95 
96 
97 
98 
99 

100 

lUl 

102 

103 

104 

105 

106~ 

107 

103 

109 

110 

111 

112 

113 

114 

115 

116 

117 

118 

119 

120 
121 
122 
123 
124 


3969 
4016 
4225 
4356 
4439 
4624 
4761 

4900 
504 1 
5184 
5329 
5476 
5025 
57  76 
5929 
6034 
6241 

6400 
6561 
6724 
6339 
7056 
7225 
7396 
7569 
7744 
7921 

8100 
8231 
8464 
8619 
8336 
9025 
9216 
9409 
96:)4 

10000 
10201 
10404 
10609 
10316 
11025 
11236 
11449 
11664 
1 1331 

12100 

12321 

12544 

12769 

12996 

13225 

13456 

13639 

13924 

14161 

14400 
14611 
14334 
15129 
15376 


250047 
262144 
274625 
2o7496 
300763 
314432 
323509 

343000 
357911 
373248 
339017 
405224 
421375 
433976 
456533 
474552 
493039 

512000 
531441 
551368 
57 1 737 
592704 
614125 
636056 
653503 
631472 
704969 

729000 

i  OOOi I 

773633 
804357 
830534 
857375 
834736 
912673 
911192 
970299 

1000000 
1030301 
1061203 
1092727 
H24.S64 
1157625 
1191016 
1225043 
1259712 
1295029 

1331000 

1367631 

1404928 

1442897 

1431 544 

1520375 

1560896 

1601613 

1613032 

1635159 

1723000 
1771561 
1815848 
1860367 
1906624 


7.9372539 
8.0000000 
8.0622577 
8.1240334 
8.1853528 
8.2462113 
8.3066239 

8.3666003 

8.4261498 
8.4352314 
3.54 10U37 
8.6023253 
8.6602S40 
8.7177979 
8.7749644 
8.831 7  609 
8.8831944 

8.9442719 
9.0000000 
9.05.53351 
9.1104336 
9.1651514 
9.2195445 
9.2733185 
9.3273791 
9.3303315 
9.4339311 

9.4363330 
9.5393920 
9..59 10630 
9.6436508 
9.6953597 
9.7467943 
9.7979590 
9.8483578 
9.8994949 
9.949o744 

lo.onooooo 

10.0493756 
10.0995049 
10.1433916 
10.1930390 
10.2469508 
10.2956301 
i0.34i'h04 
10.3923048 
10.4403065 

10.4330035 
10.5356538 
10.5330052 
10.6301458 
10.67707S3 
10.7233053 
1 0.7703296 
10.8166538 
10.8627805 
10.9087121 

10.9544512 
11.0000000 
11.0453610 
11.0905365 
11.1355237 


3.9790571 

4.0000000 ■ 

4.02()72.'6 

4.0112401 

4.0615130 

4.0816551 

4.1015661 

4.121 23.'i3 
4.14U3173 
4,1601676 
4.1793390 
4.1933364 
4.2171633 
4.2358236 
4.2543210 
4.2726536 
4.2903404 

4.3083695 
4.32674b7 
4.3444815 
4.3620707 
4.3795191 
4.3963296 
4.4140049 
4.4310476 
4.4479602 
4.4647451 

4.4814047 
4.4979414 
4.5143574 
4.5306549 
4.54633.59 
4.5629026 
4.5738570 
4.5947009 
4.6104363 
4.6260650 

4.6415888 
4.6570095 
4.6723287 
4.6375482 
4.70^6694 
4.7176940 
4.7326235 
4.7474594 
4.7622032 
4.7763562 

4.7914199 

4.895t955 
4.3202345 
4.8345331 
4.848^076 
4.8629442 
4.8769990 
4.8909732 
4.9043631 
4.9186347 

4.9324242 
4.9460374 
4.9596757 
4.9731898 
4.9366310 


.015873016 
.015625000 
,015384615 
.015151515 
.014925373 
.014705332 
.0144927.54 

0142N5714 
.0140^4507 
.(0  38^3889 
.01369.'-630 
.013513514 
.013333333 
.013157895 
.012987013 
.012820513 
.012653223 

.012500000 
.012345679 
.012195122 
.012048193 
.011904762 
.011764706 
.011627907 
.011494253 
.011363636 
.011235955 

.011111111 

.010939011 
.010^69565 
.010752638 
.01063.>298 
.010526316 
.010416667 
.010309278 
.0102(14032 
.010101010 

.010000000 
.009900990 
.009i03922 
.00970i-738 
.001,61.53'?5 
.009523810 
.009433962 
.009345794 
.009259259 
.009174312 

.O090909'i9 
.009009009 
.00>:923571 
.003349.5.58 
.Olte771930 
.003695652 
.003620690 
.003547009 
.003474576 
.008403361 

.008333333 
.003261463 
.008196721 
.008130081 
.008061516 


14U 

TABLE    XI.        SQUARES,    CCBES, 

SQUARE    ROOTS, 

No. 

Squares. 

Cubes 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

125 

1.5625 

1953125 

11.1303399 

5.0000000 

.O030JO000 

126 

15S76 

2tr))376 

11.2249722 

5  0132979 

.007936503 

127 

16129 

2)43333 

11.2694277 

5  0265257 

.007874016 

12S 

163S4 

2)97152 

11.3137035 

5.0396342 

.007812;500 

12J 

16611 

21463^9 

11.3573167 

5.0527743 

.0077519.33 

133 

16900 

21 970 JO 

11.4017543 

5.0657970 

.007692303 

131 

17161 

2243091 

11.4155231 

5.0737.531 

0)7633533 

132 

17424 

2299963 

11.4391253 

5.0916134 

.01)757.5753 

133 

176S9 

2352637 

11. .532.5626 

5.10446^7 

.007513797 

i;i4 

179.56 

2106104 

11.5753369 

5  1172299 

.0)7462637       ; 

13.5 

18225 

2460375 

11.6139.500 

5  1299273 

.037407407 

136 

13496 

251.54.56 

11.6619033 

5  1425632 

.0073.52941 

137 

13769 

2.571353 

11.7046999 

5.1-551367 

.0[)7299270 

13S 

19044 

2623072 

11.7473444 

5  167^193 

.007246377 

139 

19321 

263.5619 

11.7393261 

5.1301015 

.007194245 

140 

19600 

2744010 

11.3.321596 

5.1924941 

.007142357       i 

141 

19331 

23f)322[ 

1 1.3743121 

5.2043279 

.007092199       1 

142 

20161 

2363233 

11.916.3753 

5  2171034 

.0070422^54       ; 

143 

20449 

2921207 

11.9532607 

5.2293215 

.006993007 

144 

20736 

2935934 

12.0'JOOOOO 

5.2414S23 

.006944444 

14o 

21 025 

.3013625 

12.041.5946 

5.2.535379 

.006>96552 

146 

21316 

3II2136' 

12.0330460 

5.2656374 

.0OG349315 

147 

21609 

3176523 

12.1243557 

5.2776.321 

.00630272         i 

U3 

21904 

32  U  792 

12.16-55251 

5.2395725 

.00075675?       ' 

149 

22201 

3307949 

12.2065558 

5..3014592 

.00671 1409       1 

150 

22500 

3375000 

12  2474437 

5.31.32923 

.006666667 

151 

22301 

3142951 

12  23320.57 

5.3250740 

.0)6622517 

1         152 

23101 

.3511303 

12.3233230 

5  3363033 

.006573947       | 

153 

23109 

3531577 

12  3693169 

5.3434312 

.006535943 

154 

23716 

36.52264 

12.4096736 

5..3601034 

.006493506 

155 

24025 

3723375 

12.4493996 

5.37163.54 

.006451613 

156 

21.3.36 

3796416 

12.4509960 

5.3332126 

.006410256 

157 

21619 

3369393 

12..5293641 

5.3946207 

.006369427 

15S 

24961 

3944312 

12..5693051 

5.4051202 

.006329114 

159 

252S1 

4019679 

12.6095202 

5.4175015 

.006239303 

160 

25600 

4096000 

12.6491106 

5.4233-3^52 

.006250000 

161 

2.5921 

4173231 

12.6335775 

.5.4401218 

.006211130 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172-40 

163 

26.569 

4330747 

12.7671453 

5.46255.56 

.006134969 

164 

26596 

•4410944 

12.3062435 

5.4737037 

.006097561 

165 

27225 

4492125 

12.3452326 

5.4343066 

.006)60606 

166 

27556 

4574296 

12.3310937 

5.49-53647 

.006024096 

167 

27339 

4657463 

12.9223430 

5.5063784 

.01.5933024      i; 

163 

23221 

4741632 

12.9614314 

5.5173434 

.00-59.52331      j 

169 

23561 

4326309 

13.0000000 

5..5237748 

.035917160 

170 

23900 

4913000 

13.0334043 

.5.5396533 

.005332.353 

171 

29241 

5000211 

I.3.0r66963 

5.5.504991 

.00.53479-53 

172 

29534 

5033443 

13.1143770 

5.5612973 

.00.53139.53 

173 

29929 

5177717 

13.1529464 

5.5720.546 

.005730347 

174 

30276 

5263024 

13.1909060 

5.5327702 

.00.5747126 

175 

30625 

5359375 

13.2237566 

5..5934447 

.005714236 

176 

30976 

5451776 

13.2661992 

5.6040737 

.00.5631318 

177 

31329 

.5545233 

13.3011347 

5.6146724 

.005649713 

178 

316S4 

5639752 

13.3416641 

5.6252263 

.00.561797.^ 

179 

32)41 

573.5339 

13.3790332 

5.6357403 

.005536592 

130 

32400 

.5332000 

13  4164079 

5.6462162 

.005555-556 

181 

32761 

5929741 

13.4536240 

5.6566.523 

.005524362 

132 

33124 

6023563 

13.4907376 

5.6G705!] 

005494505 

133 

.33439 

6123437 

13.5277493 

5.6774114 

.005161431 

134 

.333.56 

6229.504 

13.5646600 

5.6377:340 

.005434733 

13.5 

34225 

6331625 

13.6014705 

5.6930192 

.ftO54O.540E 

186 

34596 

d434356 

1.3.6331317 

5.7032675 

.005376344 

CUBE    KOOTS,    AND    RECII EOCALS. 


141 


T- 


No. 

1S7 

158 
ISO 

190 
191 
192 
193 
194 
195 
196 
197 
198 
199 


210 
211 
212 
213 
214 
215 
216 
217 
213 
219 

220 
221 

222 
22-3 
224 
225 

226 
227 
223 
229 


240 
241 
242 
243 
244 
245 
246 
247 
243 


Squares. 

■'A'.  69 
:i5:{l4 
35721 

36100 
;-i(.4-i 
36364 
37249 
37636 
38025 
38416 
38S09 
39204 
39601 


200 

40000 

201 

40401 

202 

40304 

203 

41209 

204 

41616 

205 

42025 

206 

42436 

207 

42349 

203 

43264 

209 

43631 

Cubes. 


Square  Roots. 


41100 
44521 
44944 
45369 
45796 
46225 
46656 
47039 
47524 
47961 

48400 
43841 
49284 
49729 
50176 
50625 
51076 
51529 
5 1934 
52441 


230 

52900 

231 

53361 

232 

53824 

233 

54289 

234 

54756 

235 

55225 

236 

55696 

237 

56169 

238 

56644 

2.39 

57121 

57600 
58081 
5S5M 
59049 
59536 
60025 
60516 
61009 
61504 


6539203 
6644672 
6751269 

6859000 
6967871 
7077838 
7139057 
7301384 
7414875 
7529536 
7645373 
7762392 
7330599 

8000000 
8120601 
8242408 
836542? 
8489664 
8615125 
8741816 
8369743 
8993912 
9129329 

9261000 

9393931 

9523128 

9663597 

9S00;344 

9933375 

10077696 

1021S313 

10360232 

10503459 

10648000 
10793361 
10941048 
11039567 
11239424 
11390625 
11543176 
11897083 
11852352 
12008989 

12167000 
1232S391 
1 2437 1 68 
12649337 
12812904 
12977875 
13144256 
13312053 
13431272 
13651919 

13324000 
13997521 
14172438 
14343907 
14526784 
14706125 
14336936 
15069223 
15252992 


Cube  Roots. 


13  6747943 
13.7113092 
13.7477271 

1.3.7840138 
13.8202750 
13.8564065 
13.8924440 
13.9233383 
13.9642400 
14.0000000 
14.03.56688 
14.0712473 
14.1067360 

14.1421356 
14.1774469 
14.2126704 
14.2473068 
14.2828569 
14.3173211 
14.3527001 
14.3374946 
14.4222051 
14.4568323 

14.4913767 
14..5258390 
14.5602198 
14.5945195 
14.6237388 
14.6623733 
14.6969385 
14.7309199 
14.7643231 
14.7986488 

14.8323970 
]4.866(i637 
14.8996644 
14.9331345 
14.9666295 
15.0000000 
15.0332964 
15.0665192 
15.0996639 
15.1327460 

15.1657509 
15.1936342 
15.2315462 
15.2643375 
15.2970535 
15.3297097 
15,3622915 
15,3943043 
15,4272486 
15.4596248 

15.49193.34 
15.5241747 
1.5.5563492 
15..58S4573 
15.6204994 
15.6524758 
15.6343371 
15,7162336 
15.7480157 


Reciprocalfi. 


5.71S4791 
5.7236543 
5.7387936 

5,7438971 
5,7539652 
5.7639932 
5.7739966 
5.7889604 
5.7983900 
5  8037857 
5.8136479 
5.8284767 
5.3382725 

5.8480355 
5.8577660 
5.3674643 
5.8771307 
5,83676.53 
5.8963685 

5  9059406 
5.91.54817 
5.9249921 
5.9344721 

5.9439220 
5.953.3418 

5.9627320 
5.9720926 
5.9314240 
5.9907264 
6.0000000 
6.00924.50 
6.0184617 
6.0276502 

6.0368107 
6,0459435 

6  0550489 
6.0641270 
6.0731779 
6.0822020 
6.0911994 
6.1001702 
6.1091147 
6.1180332 

6.12692.57 
6.1357924 
6.1446337 
6.1534495 
6.1622401 
6.1710058 
6.1797466 
6,1884628 
6.1971544 
6.20.58218 

6,2144050 
6.2230843 
6.2316797 
6,2402515 
6.2487998 
6.2573248 
6,2653266 
6,2743054 
6,2327613 


.005347591 
.00.5319149 
005291 U05 

.005263153 
.005235602 
.005208333 
.005181347 
.0051.54639 
.005128205 
.005102041 
.005076142 
.005050505 
005025126 

.005000000 
.004975124 
,004950495 
.004926108 
,004901961 
.004378049 
.004354369 
.004830918 
.004807692 
.004784639 

,004761905 
,0047393:-6 
.0047169S1 
.004694336 
,004672397 
.004651163 
,004629630 
.004603295 
,004587156 
.004566210 

,004545455 
.004524387 
,004504505 
,004434305 
.004464236 
,004444444 
,004424779 

35S 

;5£ 
.004366812 

.001347826 
.004329004 
.004310345 
.004291845 
.004273504 
.0042,55319 
,0CI42372S3 
,004219409 
.004201681 
.004184100 

.004166667 
.004149378 
,004132231 
.004115226 
.004098361 
.0040816.33 
004065041 
.004043583 
.004032258 


142 

TABLE  XI.   SQUARES,  CUBES, 

SQUARE  R«'ul&, 

!  -1 

No. 

Squares. 

Cubes. 

Square  Roots 

Cube  Roots. 

Reciprocals. 

249 

62001 

154.38249 

15.7797333 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113383 

6.2996053 

.004000000 

251 

53001 

15813251 

15.3429795 

6.3079935 

.00.3934064 

252 

63504 

16103003 

15.3745079 

6.3163.596 

.0039632.54 

253 

64009 

16194277 

15.9059737 

6.3247035 

.0039-52569 

251 

64516 

16387064 

15.9373775 

6.3330256 

.003937003 

255 

65025 

16581375 

15.9637194 

6.34132.57 

.003921569 

256 

65536 

16777216 

16.0000000 

6.3196042 

.00-3906250 

257 

66049 

16974593 

16.0312195 

6.3573611 

.003391051 

25S 

66564 

17173512 

16.0623734 

6.3663963 

.00387.5969 

259 

670S1 

17.37.3979 

16.0934769 

6.. 37431 11 

.003361004 

260 

67600 

17576000 

16.12451.55 

6.3325043 

.00.3346154' 

261 

68121 

17779581 

16.1.5.54944 

6.-39)6765 

.00.3831418 

26-2 

68644 

17931723 

16.1864141 

6.. 3938279 

.03.3316:5:94 

263. 

69169 

18191447 

16.2172747 

6.4069535 

.003302231 

264 

69696 

18.399744 

16.2430763 

6.41.50637 

.003787879 

265 

70225 

18609625 

16.2783206 

6.4231533 

.003773585 

266 

70756 

18321096 

16.309.5064 

6.4312276 

.003759398 

267 

71289 

19034163 

16.3401346 

6.4392767 

.00374.5318 

203 

71824 

19243332 

16.3707055 

6.4473057 

.003731.343 

269 

72361 

19165109 

16.4012195 

6.4553143 

.003717472 

270 

72900 

1938.3000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003690037 

272 

73984 

20123643 

16.4924225 

6.4792236 

.003676471 

273 

74529 

20346417 

16.5227116 

6.4371541 

.003663004 

274 

75076 

20570324 

16.0.529454 

6.49506.53 

.003649635 

275 

75625 

20796375 

16.5831240 

6.. 5029572 

.003636364 

276 

76176 

21024576 

16.6132477 

6.5103300 

.003623133 

277 

76729 

212539.33 

16.6433170 

6.5186339 

.00.3610108 

278 

77284 

21434952 

16.6733.320 

6..5265139 

.003597122 

279 

77841 

21717639 

16.7032931 

6.. 5343351 

.003534229 

230 

78400 

219.52000 

16.7332005 

6.5421326 

.00.3571429 

2S1 

73961 

22188041 

16.7630.546 

6.  .54991 16 

.003553719 

282 

79524 

22425768 

16.7923.5.56 

6.5576722 

.003.546099 

283 

80039 

22665] 37 

16.3226033 

6.. 56.54 144 

.003.5.3.3569 

284 

80656 

229(16304 

16.3-522995 

6.5731335 

.00.3521127 

285 

81225 

23149125 

16.8819430 

6.5303443 

.003503772 

286 

81796 

23393656 

16.9115.345 

6.5385323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.5962023 

.003434321 

288 

82944 

23887372 

16.970.5627 

6.6033545 

.003472222 

289 

83521 

241.37569 

17.0000300 

6.6114890 

.0034613203 

290 

84100 

24339000 

17.0293=64 

6.6I910S0 

.00.3443276 

291 

84631 

24642171 

17.0.537221 

6.62670.54 

.00:}4-36426 

292 

85264 

243970  S8 

17.0380075 

6.6342874 

.003424653 

293 

85849 

251537.57 

17.1172123 

6.6113.522 

.033412969 

294 

86436 

25112184 

17.1464232 

6.6193993 

.003401-361 

295 

87025 

25672375 

17. 175.56  to 

6.6569302 

.003339831 

296 

87616 

25931336 

17.2046.505 

6.6644437 

.003378378 

297 

38209 

26193073 

17.2.33G879 

6  6719403 

.003367003 

298 

88304 

26163.592 

17.2626765 

6.6794200 

.003355705 

299 

89101 

26730399 

17.2916165 

6.6863831 

.003344432 

300 

90000 

27000000 

17.320.5081 

6.694-3295 

.00.3333333 

301 

90601 

27270901 

17.349.3516 

6.7017593 

.003322259 

302 

91204 

27543603 

17.3781472 

6.7091729 

.0033112.58 

303 

91309 

2781S127 

17.4Q68952 

6.7165700 

.003300330 

304 

92416 

23094464 

17.435.59.53 

6.7239503 

.003289474 

305 

93025 

23372625 

17.4642492 

6.73131.55 

.00.3278639 

306 

93636 

236.52616 

17.4928557 

6.7336641 

.003267974 

307 

94249 

23934443 

17.5214155 

6.7459967 

.00.3257329 

308 

94864 

29213112 

17.&499283 

6.7.533134 

.003246753 

309 

9.5481 

29503629 

17.. 578.39.53 

6.7606143 

.003236246 

310 

96100 

29791000 

17.6063169 

6.7673995 

.00322.5806 

CUBE    ROOTS,    AND    KECIPROCALS. 


143 


No. 

311 
312 
313 
314 
315 
316 
317 
318 
319 

320 
321 
322 
323 
324 
32.5 
326 
327 
325 
329 

330 
331 
332 
333 
334 
335 
336 
337 
33S 
339 

340 
341 
342 
343 
344 
345 
346 
347 
343 
349 

350 
351 
3.52 
353 
351 
355 
3.56 
357 
358 
359 

360 
361 
362 
363 
364 
365 
366 
367 
368 
369 

370 
371 
372 


IL. 


Squares. 

96721 

97344 

97969 

9S596 

99225 

99S56 

100489 

101124 

1U1761 

102100 
103041 
10.36S1 
104329 
104976 
10.5625 
] 06276 
1(16929 
1075S4 
103241 

108900 
109561 
110224 
11 0339 
111556 
112225 
112S96 
113.569 
1 14244 
114921 

115600 
116231 
116964 
117619 
113336 
119025 
119716 
120409 
121104 
121301 

122500 
12-3201 
12.3904 
124609 
125316 
126025 
126736 
127449 
128164 
128881 

129600 
130.321 
131044 
131769 
132496 
133225 
1339.56 
134639 
13.5424 
136161 

136900 
137641 
133384 


Cubes. 


Square  Koots. 


Cube  Roots. 


30080231 
30371328 
30664297 
309.59144 
31255875 
31554496 
31855013 
321574.32 
32461759 

32763000 
33076161 
3;53S6243 
33698267 
34012224 
34323125 
34615976 
34965783 
352-57552 
3-561 1239 

35937000 
38264691 
36594363 
36926037 
37259704 
37595375 
37933056 
38272753 
336H472 
33953219 

39.304000 
39651821 
40001688 
40353807 
40707584 
4106.3625 
41421736 
41781923 
42144192 
42508549 

42375000 
4.3213551 
43614208 
439S6U77 
44361364 
44733375 
45118016 
45499293 
458S2712 
46263279 

466.56000 
47045831 
47437923 
47832147 
48228544 
48627125 
49027396 
49430S63 
49836032 
50243409 

5065.3000 
51064811 
51478S48 


17.6351921 
17.6635217 
17.6918060 
17.7200451 
17.7432393 
17.7763388 
17.8044933 
17.8325,545 
17.fc605711 

17.888.54.38 
17.9164729 
17.9443.534 
17.9722008 
18.0000000 
18.0277564 
18.0554701 
18.0.-31413 
18.1107703 
18.138.3571 

18.1659021 
18.1934054 

13.2203672 
1S.24S2876 
18.2756669 
18.3030052 
18.3303023 
18.3575598 
18.3347763 
18.4119526 

18.4390889 
18.46618.53 
18.4932420 
18.5202592 
18.5472370 
13.5741756 
18.80107.52 
18.6279360 
18.6.547581 
18.6315417 

18.7082S69 
18.7.349940 
18.7616630 
18.7882942 
13.8143877 
18.8414437 
18.8679623 
18.8944436 
18.9203879 
18.9472953 

18.9736660 
19.0000000 
19.0262976 
19.0.52.5589 
19.0787840 
19.1049732 
19.1311265 
19.1.572441 
19.1833261 
19.2093727 

19.2353841 
19.2613603 
19.2873015 


Reciprocals. 


6.7751690 
6.7324229 
6.7396613 
6.7963344 
6.8040921 
6.8112347 
6.8184620 
6.8256242 
6.8327714 

6.8399037 
6.8470213 
6.8.541240 
6.8612120 
6.8632355 
6.8753443 
6.8323888 
6.8894188 
6.8964.345 
6.9034359 

6.91042.32 
6.917.3964 
6.9243556 
6.9313008 
6.9.332321 
6.9451496 
6.9520533 
6.9589434 
6.9658198 
6.9726S26 

6.9795321 
6.9S63631 
6.99319(16 
7.0000000 
7.0067962 
7.013.5791 
7.0203490 
7.02710.58 
7.03.33497 
7.0405806 

7.0472987 
7.0.540041 
7.0606967 
7.0673767 
7.0740440 
7.0806988 
7.0873411 
7.0939709 
7.1005SS5 
7.10719.37 

7.1137866 
7.1203674 
7.1269360 
7.1334925 
7.1400370 
7.146.5695 
7.1530901 
7.1.595938 
7.1660957 
7.1725809 

7.1790544 
7. 1855 162 
7.1919663 


.003215434 
.003205128 
.003194388 
.003134713 
.003174603 
.003164557 
.003154574 
.003144654 
.003134796 

.003125000 
.003115265 
.00310.5590 
.003095975 
.003036420 
.003076923 
.003067435 
.0030.58104 
.003048780 
.003039514 

.003030303 
.003021148 
.003012048 
.003003003 
.002994012 
.002935075 
.002976190 
.002967359 
.0029585^(1 
.002949353 

.002941176 
.002932551 
.002923977 
.002915452 
.002906977 
.002898551 
.002890173 
.002381844 
.002873563 
.002865330 

.002357143 
.002849003 
.002840909 
.002832861 
.002824859 
.002816901 
.002808989 
.002301120 
.002793296 
.002785515 

.002777773 
.002770083 
.002762431 
.002754321 
.002747253 
.002739726 
.002732240 
.002724796 
.002717391 
,002710027 

.0027Lr2703 
.002695418 
.002688172 


I 

11 

TABLE  XI.   SQUARES,  CUBES, 

SQUARE  ROOTS, 

No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocal*. 

373 

139129 

51395117 

19.3132079 

7.19340.50 

.002630965 

371 

139376 

52313624 

19.. 3390796 

7.204332  a 

.002673797 

375 

140325 

52734375 

i9..3649167 

7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194 

7.2176522 

.0026.59574 

377 

142129 

53532633 

19.4164373 

7.2240450 

.0026.52.520 

373 

142334 

54010152 

19.4422221 

7.2304263 

.002645503 

379 

143641 

54439939 

19.4679223 

7.2367972 

.002633522 

330 

144400 

54372000 

19.4935337 

7.2431565 

.002631579 

331 

145161 

55306:M1 

19.5192213 

7.2495045 

.002621672 

332 

145924 

55742963 

19.5443203 

7.2.553415 

.002617301   1 

333 

146639 

56131337 

19.5703353 

7.2621675 

.002610966 

334 

147456 

56623104 

19..5959179 

7.2GS4-24 

.0026  m  67   I 

335 

143225 

57066625 

19.6214169 

7.2747^64 

.002597403 

336 

143993 

57512456 

19.6463327 

7.2310794 

.002590674 

337 

149769 

57960603 

19.67231.56 

7.2373617 

.00253:3979 

333 

150544 

53411072 

19.69771-56 

7.29.36330 

.002577320 

339 

151321 

53363369 

19.72.30329 

7.2993936 

.002570694 

390 

152100 

59319000 

19.7434177 

7.30314.36 

.002564103 

391 

152331 

59776471 

19.7737199 

7.3123^23 

.002557545 

392 

153664 

60236233 

19.7939399 

7.3136114 

.002.551020 

393 

154449 

60693457 

19.3242276 

7.-3243295 

.002.544529 

394 

155236 

61162934 

19.^494332 

7.3310369 

.002533071 

395 

156025 

61629375 

19.8746069 

7.3372339 

.002531646 

396 

156316 

62099136 

19.3997437 

7.34.34205 

.00252.52.53 

397 

157609 

62570773 

19.9243533 

7.3495966 

.002518392 

393 

153404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749344 

7.3619178 

.002506266 

400 

160000 

64090000 

20.0000000 

7.-3630630 

.002.500000 

401 

160301 

64431201 

20.0249344 

7.3741979 

.002493766 

402 

161604 

61964303 

20.0499377 

7.-3303227 

.002437562 

403 

162409 

6.5450327 

20.0743599 

7.3364373 

.002431390 

404 

163216 

65939264 

20.0997512 

7.3925418 

.00247.5243 

405 

164025 

66431125 

20.1246113 

7.-3936363 

.002469136 

406 

161S36 

66923416 

20.1494417 

7.4047206 

.00246:30.54 

407 

165649 

6741*143 

20.1742410 

7.4107950 

.002457002 

403 

166464 

67917312 

20.1990099 

7.4163595 

.0024.50980 

409 

167231 

63417929 

20.2237434 

7.4229142 

.002444938 

410 

163100 

63921000 

20.2434.567 

7.4239.539 

.002439024 

411 

163921 

69426531 

20.2731349 

7.4:349933 

.00243-3090 

412 

169744 

699:34523 

20.2977631 

7.4410139 

.002427184 

413 

170569 

70444997 

20.3224014 

7.4470-342 

.002421.303 

414 

171396 

70957944 

20.3469399 

7.4530.399 

.00241:5459 

415 

172225 

71473375 

20.371.5433 

7.4590-3.39 

.002409639 

1      ■■■*■-' 

.   416 

173056 

71991296 

20.  .3960781 

7.4650223 

.002403346 

417 

173339 

72511713 

20.4205779 

7.4709991 

.002393032 

413 

174724 

73031632 

20.4450433 

7.4769664 

.002392.344 

419 

175561 

73560059 

20.4694395 

7.4529242 

.002356635 

420 

176400 

74033003 

20.4939015 

7.4833724 

.002330952 

421 

177241 

74613461 

20.5132345 

7.4943113 

.002375297 

422 

17S034 

75151443 

20..5426336 

7.5007406 

.002369663 

423 

173929 

75636967 

20.5669633 

7.5066607 

.002364066 

424 

179776 

76225024 

20.5912603 

7.5125715 

.002-353491 

425 

139625 

76765625 

20.615.5231 

7.5134730 

.002:3.52941 

426 

181476 

77303776 

20.6397674 

7.5243652 

.002-347418 

427 

132329 

77S;54433 

20.66397S3 

7.5302432 

.002-341920 

423 

133134 

78402752 

20.6331609 

7.5361221 

.002336449 

429 

184041 

78953539 

20.71231.52 

7.5419367 

.002-331002 

430 

184900 

79507000 

20.7.364414 

7.5473423 

.002325581 

431 

185761 

80062991 

20.7605395 

7.55.36333 

.002:320186 

432 

1S6624 

80621563 

20.7846097 

7.5595263 

.002314315 

433 

137439 

81132737 

20.8036.520 

7.5653.543 

.002.309469 

434 

1S3356 

81746504 

20.8326667 

7.5711743 

1  .002.304147 

CUBE    ROOTS,    AM>    R  KCll'ROCALS. 


145 


No. 

43.3 
4:3fi 

•137 
■i.-'S 
439 

410 
4-;i 
442 
413 
444 
445 
440 
447 
44S 
449 

.  450 
451 
4.52 
453 
4.54 
455 
456 
457 
45S 
459 

460 
461 
462 
463 
46! 
465 
466 
4/57 
463 
469 

470 
471 
472 
473 
474 
475 
476 
477 
478 
479 

4S0 
481 
482 
483 
484 
435 
4  So 
487 
488 
439 

490 
491 
492 
493 
494 
495 
496 


\... 


Squares. 

189225 
1 9U0.;6 
1  '.K)ii69 
191844 
192721 

193600 
194481 
195364 
196249 
197136 
19-025 
19^916 
I 99S09 
200704 
201601 

202500 
203101 
2043111 
205209 
2061 16 
207025 
207936 
208849 
209764 
210681 

211600 
212521 
21.3444 
214369 
215296 
216225 
217156 
218089 
219024 
219961 

220900 
221841 
222784 
223729 
224676 
225625 
226576 
227529 
228484 
229441 

2.30400 
231361 
232324 
233289 
234256 
235225 
236196 
237169 
233144 
239121 

240100 
241081 
242061 
243049 
244036 
245025 
246016 


Cubes 


Square  Roots. 


82312875 
82881856 
83  i^' 3453 
Sl(l:;7672 
f46('4519 

S51e84000 
85766121 
86350888 
8693;307 
S752>^3>4 
88121125 
88716536 
89314623 
89915392 
90518849 

9112.5000 
91733851 
92345403 
92959677 
93576664 
94196375 
948 188 16 
95443993 
96071912 
S6702579 

97336000 

97972181 

98611128 

99252847 

99897344 

100.544625 

101194096 

101847.563 

102503232 

103101709 

103823000 
104487111 
105154048 
10.5823S17 
106496424 
107171875 
1078.50176 
1035313.33 
10921.53.52 
109902239 

110592000 
1H2S4641 
111930168 
112678.587 
113379904 
1140-^4125 
114791256 
11.5501303 
116214272 
116930169 

117649000 
118370771 
119095488 
119^23157 
120553784 
121287375 
12202.3936 


Cube  Roots. 


20.8566536 
20.8806130 
20.904.5450 
20.9284495 
20.9523263 

20.9761770 
21.0000000 
21.0237960 
21.0475652 
21.0713075 
21.0950231 
21.1187121 
21.142.3745 
21.1660105 
21.1896201 

21.2132034 
21.2367606 
21.2602910 
21.2837967 
21.3072758 
21.3307290 
21.3541565 
21.3775583 
21.4009346 
21.4242853 

21.4476106 
21.4709106 
21.4941853 
21.5174348 
21.5406592 
21.. 5638.587 
21.5870331 
21.6101828 
21.6333077 
21.0564078 

21.0794834 

21.7025344 

21.72.55610 

21.748,5632 

21.7715411 

21.7944947 

21.8174242- 

21.8403297 

21.8632111 

21.8500636 

21.9089023 
21.9317122 
21.9544984 
21.9772610 
22.0000000 
22.02271.55 
22.04.54077 
22.0680765 
22.0907220 
22.11.33444 

22.13594.36 
22.1.585193 
22.1810730 
22.2030033 
22.2261103 
22.2485955 
22.2710575 


Reciprocals. 


7.5769849 
7.5827865 
7.5885793 
7.5943633 
7.6001385 

7.60.59049 
7.6116626 
7.6174116 
7.6231519 
7.6288837 
7.6346067 
7.6403213 
7.0460272 
7.6517247 
7.0574138 

7.6630943 
7.6687665 
7.6744303 

7.6800857 
7.6857.323 
7.6913717 
7.6970023 
7.7026246 
7.7082388 
7.7138448 

7.7194426 
7.7250325 
7.7306141 
7.7361877 
7.7417532 
7.7473109 
7.7523606 
7.7584023 
7.7639261 
7.7694620 

7.7749301 
7.7804904 
7.7859928 
7!  79 14875 
7.7909745 
7.80245.38 
7.80792.54 
7.8133392 
7.8188456 
7.8242942 

7.8297353 
7.8351638 
7.8405949 
7.8460134 
7.8514244 
7.8568281 
7.8622242 
7.8676130 
7.8729944 
7.S7830S4 

7.8837352 
7.8890940 
7.8944403 
7.8997917 
7.9051294 
7.9104.599 
7.9157832 


.002298851 
.002293578 
002288330 
.002283105 
.002277904 

.002272727 
.002267574 
.002262443 
.002257330 
.002252252 
.002247191 
.002242152 
.002237130 
.0022.32143 
.002227171 

.002222222 
.002217285 
.002212389 
.002207506 
.002202643 
.002197802 
.002192982 
.002188184 
.00218.3406 
.002178649 

.00217.3913 
.002109197 
.002164502 
.002159827 
.002155172 
.002150538 
.00214.5923 
.002141328 
.0021.30752 
.002132196 

.002127660 
.008123142 
.002118644 
.002114165 

.002109705 
.002105263 
.002100840 
.002096436 
.002092050 
.002087633 

.002083333 
.002079002 
.002074689 
.002070393 
.002066116 
.0020618.56 
.002057613 
.002053388 
.002049180 
.002044990 

.002040816 
.002036660 
.002032520 
.002028398 
.002024291 
.002020202 
.002(00129 


4(5 

TABLE  Xf.   SQUARES,  CUBES, 

SQUARE  ROOTS, 

No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocal*. 

497 

247009 

122763473 

22.29:34963 

7.9210994 

.002012072 

493 

243001 

123505992 

22.3159136 

7.9264035 

,002008032 

499 

249001 

124251499 

22.3333079 

7.9317104 

.002004003 

sao 

250000 

125000000 

22.3606793 

7.9370053 

.002000000 

501 

2510J1 

125751501 

22.3330293 

7.9422931 

.001996003 

502 

252004 

126506033 

22.4053565 

7.9475739 

.001992032 

533 

253009 

127253527 

22.4276615 

7.9.523477 

.001938072 

504 

254016 

12S024064 

22.4499443 

7.9581144 

.001934127 

505 

255025 

123787625 

22.4722051 

7.9633743 

.001930193 

506 

256036 

129554216 

22.4944433 

7.9636271 

.001976285 

507 

257049 

130323343 

22.5166605 

7.9733731 

.001972337 

503 

253064 

131096512 

22.5333553 

7.9791122 

.001963534 

509 

259031 

131372229 

22.5610233 

7.9343444 

.001964637 

510 

260100 

132651030 

22.  .533 1796 

7.9395697 

.031960734 

511 

261121 

133432331 

22.6353091 

7.9947833 

.001956947 

512 

262144 

134217723 

22.6274170 

8.0300000 

.0019.53125 

513 

263169 

135005697 

22.6495033 

8.0352049 

.001949318 

514 

264196 

135796744 

22.671.5631 

8.0104032 

.00194.5525 

515 

265225 

136590375 

22.6936114 

8.0155946 

.001941748 

518 

266256 

137333096 

22.71.563.34 

8.0207794 

.001937934 

517 

267239 

133133413 

22.7376340 

8.0259574 

.001934236 

518 

263324 

133991 S32 

22.75961.34 

8.0311287 

.001930502 

519 

269361 

139793359 

22.7815715 

8.0362935 

,001926732 

520 

270400 

140603030 

22.8035035 

8.0414515 

.00192.3077 

521 

271441 

141420761 

22.82:54244 

8.0466030 

.001919386 

522 

272434 

142236643 

22.8473193 

8.0517479 

.001915709 

523 

273529 

143)55667 

22.8691933 

8.0568862 

.001912046 

5^4 

274576 

143377324 

22.8910463 

8.0620180 

.001903397 

525 

275625 

144703125 

22.9123785 

8.06714.32 

.001904762 

526 

276676 

145531576 

22.9346399 

8.0722620 

.001901141 

527 

277729 

1463631  S3 

22.9564806 

8.0773743 

.0018975.33 

523 

27S734 

147197952 

22.9732506 

8.0324300 

.001893939 

529 

279S41 

143035S39 

23.0000000 

8.0375794 

.001390359 

530 

230900 

148377000 

23.0217239 

8.0926723 

.001386792 

531 

231961 

W 972 1291 

23.0434372 

8.0977.539 

.0013332.39 

532 

233024 

150563763 

23.0651252 

8. 1023390 

.001379699 

533 

234039 

151419437 

23.0867923 

8.1079123 

.001876173 

534 

235156 

152273304 

23.1034400 

8.1129303 

.031872659 

535 

236225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

237296 

153990656 

23.15167.33 

8.12.30962 

.00186.5672 

537 

23S369 

154354153 

23.1732605 

8.1281447 

.001362197 

533 

239444 

155720372 

23.194-270 

8.1331370 

.001353736 

539 

290521 

156590319 

23.2163735 

8.1332230 

.001855288 

540 

291600 

157461000 

23.2379031 

8.14.32.529 

.001851852 

541 

292631 

153340421 

23.2594067 

8.1432765 

.00184*429 

542 

293764 

159223333 

23.2303935 

8.1532939 

.001845018 

543 

294349 

160103007 

23.30236134 

8.1533051 

.031841621 

544 

295936 

163939134 

23.3233076 

8.1633102 

.001833235 

545 

297025 

161378625 

23.3452351 

8.1633092 

.001834362 

546 

293116 

162771336 

23.3666429 

8.17.33020 

.001831502 

547 

299209 

163667323 

23.33S0311 

8.17S2833 

.0018231.54 

543 

300304 

164566592 

23.4093993 

8.1832695 

.001324818 

549 

301401 

165469149 

23.4307490 

8.1382441 

.001821494 

550 

302500 

166375000 

23.4520733 

8.1932127 

.001818182 

551 

303601 

167234151 

23.4733392 

8.1931753 

.001814832 

552 

304704 

163196603 

23.4946302 

8.2031319 

.001811594 

553 

305309 

169112377 

23.5159520 

8.2030325 

.001303318 

554 

306916 

170031464 

23.5372348 

8.21.30271 

.00180.5054 

555 

303025 

170953375 

23..55S4330 

8.2179657 

.001801302 

556 

309136 

171879616 

23.5796.522 

8.2223935 

.001793.561 

557 

310249 

172303693 

23.6033474 

8.2278254 

.031795.3.32 

553 

311364 

173741112 

23.6220236 

8.2327463 

.001792115 

CUBE    ROOTS,    AMD    RECIPROCALS. 


Ul 


No. 

559 

560 
561 
562 
563 
564 
565 
566 
567 
563 
569 

570 
571 
572 
573 

574 
575 
576 

577 
578 
579 

580 
581 
582 
5S3 
534 
585 
586 
587 
588 
539 

590 
591 
592 
593 
594 
595 
596 
597 
593 
599 

600 
601 
602 
603 
604 
605 
606 
607 
608 
609 

610 
611 
612 
613 
614 
615 
616 
617 
618 
619 
620 


Squares. 

312481 

313600 
314721 
315844 
316969 
318096 
319225 
320356 
321439 
322624 
323761 

324900 

326)11 

327184 

32S329 

32^)476 

330625 

331776 

332929 

334084 

335241 

336400 
337561 
338724 
339339 
341056 
342225 
343396 
344569 
345744 
346921 

aisioo 

349231 
350464 
351619 
352836 
354025 
355216 
356409 
357604 
358801 

360000 
361201 
362404 
363609 
364316 
366025 
367236 
363449 
369664 
370S81 

372100 
373321 
374544 
375769 
376996 
37^225 
379456 
3306S9 
331924 
333161 
3S4400 


Cubes 


Square  Roots. 


174676879 

175616000 
176558481 
177504328 
178453547 
179406144 
180362125 
181321496 
182234263 
183250432 
1842200JJ 

185193000 
1S6169411 
187149248 
183132517 
189119224 
190109375 
191102976 
192100033 
1 93 1 00552 
194104539 

195112000 
196122941 
197137368 
193155237 
199176704 
200201625 
201230056 
202262003 
203297472 
204336469 

205379000 
206425071 
207474638 
203527357 
209534534 
210644875 
21170S736 
212776173 
213.347192 
214921799 

216000000 
217031801 
218167208 
219256227 
220343864 
221-445125 
222545016 
223643543 
224755712 
225366529 

226931000 
225099131 
229220923 
230346397 
23147.5544 
232608375 
233744396 
234335113 
236029032 
237176659 
23S32S000 


Cube  Roots. 


23.6431803 

23.6643191 
23.6854336 
23.7065392 
23.7276210 
23.7486842 
23.7697236 
23.79117545 
23.8117618 
23.8327506 
23.8537209 

23.8746723 
23.8956063 
23.9165215 
23.9374184 
23.9582971 
23.9791576 
24.0000000 
24.0208243 
24.0416306 
24.0624183 

24.0S31S91 
24.1039416 
24.1246762 
24.1453929 
24.1660919 
24.IS67732 
24.2074369 
24.2230329 
24.2487113 
24.2693222 

24.2S991.56 
24.3104916 
24.3310501 
24.3515913 
24.3721152 
24.3926213 
24.4131112 
24.4335334 
24.4540335 
24.4744765 

24.4943974 
24.5153013 
24.5.3.56383 
24.5560583 
24.5764115 
24.5967473 
24.6170673 
24.6.373700 
24.6576560 
24.67792.54 

24.6981781 
24.7184142 
24.7386333 
24.7.583363 
24.7790234 
24.7991935 
24.3193473 
24.3.394347 
24.3596058 
24.8797106 
24.3997992 


Reciprocals. 


8.2.376614 

S.  2425706 
8.247474') 
8.2523715 
8.25726.33 
8.2621492 
8.2670594 
8.2719039 
8.2767726 
8.2316355 
8.2S64928 

8.2913444 
8.2961903 
8.3010304 
8.3053651 
8.3106941 
8.3155175 
8.3203353 
8.3251475 
8.3299542 
8.3347553 

8.3395509 
8.344-3410 
8.^4912.56 
8.3539047 
8.3536734 
8.3634466 
8.3632095 
8.3729668 
8.3777188 
8.3324653 

8.3372065 
8.3919423 
8.3966729 
8.4013981 
8.4061180 
8.4103326 
8.4155419 
8.4202460 
8.4249448 
8.4296333 

8.4343267 
8.4390098 
8.4436377 
8.4433605 
8.4530231 
8.4576906 
8.4623479 
8.4670001 
8.4716471 
8.4762892 

8.4309261 
8.4355579 
S.4901S43 
8.4943065 
8.4994233 
8.5ai0.350 
8.5086417 
8.5132435 
8.5173403 
8.-5224321 
8.5270139 


.001783909 

.001785714 
.001732531 
.001779359 
.001776199 
.001773050 
.001769912 
.001766784 
.001763663 
.001760563 
.001757469 

.001754336 
.001751313 
.001748252 
.001745201 
.001742160 
.0017391-30 
.001736111 
.001733102 
.001730104 
.001727116 

.001724138 
.001721170 
.001718213 
.001715266 
.001712329 
.001709402 
.001706135 
.001703.578 
.001700630 
.001697793 

.001694915 
.001692047 
.001639189 
.001636.341 
.001633502 
.001630672 
.001677852 
.001675042 
.001672241 
.001669449 

.001666667 
.001663394 
.001661130 
.0016.53375 
.001655629 
.001652393 
.0016-50165 
.001647446 
.001644737 
.001642036 

.001639344 
.0016.36661 
.001633987 
.001631321 
.001623664 
.001626016 
.001623377 
.001620746 
.001613123 
.001615509 
.001612303 


L4W 

TABLE  XI.   SQUARES,  CUBES, 

SQUARE  ROOTS, 

No. 

Squares. 

Cubes. 

Square  Roots 

Cube  Roots. 

1 

Reciprocal*. 

621 

3 356 11 

239433061 

24.9193716 

8.5316309 

.031610306 

622 

336334 

240641343 

24.9399273 

8. 5361 730 

.031607717 

623 

333129 

241304367 

24.9599679 

8.5407501 

.001605136 

624 

339376 

242970624 

24.9799920 

8.  .54531 73 

.001602564 

625 

39J625 

244140625 

25.0330000 

8.5493797 

.001600000 

626 

391376 

245314376 

25.0199920 

8.5.544372 

.001597444 

627 

393129 

246491333 

25.0.399631 

8.-5539399 

.001.594396 

62S 

394334 

217673152 

25.0599232 

8.563.5377 

.001592:357 

629 

395641 

243353139 

25.0793724 

8.5633307 

.001.539325 

630 

396903 

250047000 

25.0993003 

8. -5726 139 

.001537302 

631 

393161 

251239591 

2.5.1197134 

8.577152:3 

.001.534736 

632 

399424 

252435963 

25.1.396102 

8-5316309 

.001.532273 

633 

400639 

253636137 

25.1594913 

8.5362047 

.001.579779 

634 

401956 

254340104 

25.1793566 

8.59t)7233 

.0)1.577237 

635 

403225 

256047375 

2.5.1992063 

8.5952330 

.001574303 

636 

404495 

257259456 

25.2193404 

8.5997476 

.001-572327 

637 

405769 

253474353 

25.2333539 

8.6342.525 

.001.5693-59 

63  S 

407044 

259694072 

25.2536619 

8.6037526 

.001.567:393 

639 

403321 

■260917119 

2-5.2734493 

8.61.32430 

.001.564945 

61) 

409690 

262144030 

25.2932213 

8.6177333 

.001-562-500 

641 

410331 

253374721 

25.3179773 

8.6222243 

.001560062 

612 

412164 

264639233 

25.3377139 

8.6267063 

.0015-576:32 

613 

413149 

265347707 

25.3574447 

8.6311330 

.001-5-5-5210 

614 

414736 

267039934 

25.3771551 

8.6-3.56551 

.001-552795 

615 

416925 

263336125 

25.3963502 

8.6401226 

.001-553333 

646 

417316 

269536136 

2.5.4165301 

8.6445355 

.001547938 

647 

413609 

270340023 

25.4361947 

8.6493437 

.001-545595 

643 

419904 

272097792 

2.5.4553441 

8.65:34974 

.00154-3210 

649 

421201 

273359449 

25.47.54734 

8.6579465 

.00]  540-^.32 

650 

422503 

274625000 

25.4950976 

8.6623911 

.0015:33462 

651 

423301 

275394451 

25.5147016 

8.6663310 

.0015:36093 

652 

425104 

277167303 

25.5342907 

8.6712665 

.00l5:-;3742 

653 

426409 

273445077 

25.5533647 

8.G756974 

.001.531394 

654 

427716 

279726264 

25.5734237 

8.63012:37 

.001529(152 

655 

429025 

231011375 

25.-5929673 

8.634;54.56 

.001526713 

656 

433336 

232300416 

25.6121969 

8.6339633 

.001.524.390 

657 

431649 

233593393 

25.632flll2 

8.693:3759 

.001-522070 

653 

432951 

234390312 

2.5.6515107 

8.6977343 

.031519757 

659 

434231 

236191179 

25.6709953 

8.7021332 

.001517451 

660 

435630 

237496000 

25.69346-52 

8.7065377 

.001515152 

661 

436921 

233304731 

25.7099203 

8.7109327 

.001512359 

662 

433244 

293117523 

25.729.3607 

8.71.5-37^4 

.001510574 

663 

439569 

291431247 

25.7437364 

8.7197596 

.001503296 

664 

440396 

292754944 

25.7631975 

8.7241414 

.001.506024 

655 

442225 

294079625 

25.73759-39 

8.7235187 

.001503759 

666 

443556 

295403298 

25.3069753 

8.7.323913 

.031501502 

657 

444399 

296740963 

25.8263431 

8.7372604 

.0014992.50 

66S 

446224 

293077632 

25.3456960 

8.7416246 

.001497006 

669 

447561 

299413309 

25.3650343 

8.7459346 

.001494763 

670 

443903 

300763000 

25.8343-532 

8.7503401 

.001492537  \ 

671 

450241 

302111711 

25.9036677 

8.7.546913 

.001490313 

672 

451534 

3)3164443 

25.9229623 

8.7.590333 

.001433095 

673 

452929 

304321217 

25.9422435 

8.76-33309 

.001435334 

674 

454276 

306132024 

2.5.9615100 

8-7677192 

.001433630 

675 

455625 

307546375 

25.9307621 

8.7720532 

.001431431 

676 

456976 

303915776 

26.0300300 

8.77633.30 

.001479290 

677 

453329 

310233733 

26.0192237 

8.7307034 

.001477105 

673 

4596 34 

311665752 

26.03S433I 

8.73-50296 

.001474926 

679 

461041 

313346339 

26.0576234 

8.789-3466 

.001472754 

630 

462400 

314432000 

26.0763096 

8.7936.593 

.001470.533 

631 

453761 

315321241 

26.09-59767 

8.7979679 

.03146^129 

632 

465124 

317214563 

26.1151297 
—  ' — 

8.3022721 

.001466276 

CUBE    ROOTS,    AND    IIECIPROCALS. 


149 


No. 

6-3 
6-4 
6  So 
6S6 
6S7 
6S3 
659 

690 

691 

692 

693 

694 

695 

696 

697 

69S 

699 

700 
701 
702 
703 
704 
705 
706 
707 
703 
709 

710 
711 
712 
713 
714 
715 
716 
717 
71S 
719 


IL 


720 

721 

722 

723 

724 

725 

726 

727 

723 

729 

730 

731 

732 

733 

734 

735 

736 

737 

733 

739 

740 
741 
742 
743 
744 


Squares. 

4G64S9 
467S.36 
469225 
470596 
471969 
473344 
474721 

476100 
477431 
473S64 
4S0249 
4S1636 
4S3025 
434416 
4S5309 
437204 
433601 

490001 
491401 
492304 
4942'i9 
495GI6 
497025 
49S436 
499S49 
501264 
502631 

504100 
505521 
506944 
503369 
509796 
511 225 
512656 
514039 
515524 
516961 


Cubes. 


Square  Roots.'  Cube  Roots.   Reciprocals. 


513 100 
519341 
521234 
522729 
524176 
525625 
527076 
523529 
529934 
531441 

532900 
534361 
535324 
5372S9 
533756 
540225 
541696 
543169 
544614 
5-16121 

547600 
549)31 
550564 
552049 
553536 


31S611937 
320013504 
321419125 
322323356 
324242703 
325660672 
327032769 

323509000 
329939371 
331373333 
332312557 
334255334 
335702375 
337153536 
33S60S373 
340063392 
341532099 

343000000 
344472101 
34594340S 
34742-927 
343913664 
350402625 
351-95316 
353393243 
354394912 
356400329 

357911000 
359425431 
360944123 
362467097 
363994344 
365525375 
367061696 
363601313 
370146232 
371694959 

373245000 
374305361 
376367048 
377933067 
379503424 
331073125 
332657176 
354240533 
33532,3352 
337420439 

3^9017000 

390617591 

392223163 

393532537 

395446904 

397065375 

39563,3256 

400315553 

401947272 

403533419 

405224000 
406>69021 
40>5134S8 
41(»172407 
411530734 


26.1342687 
26.15:3.3937 
26.1725047 
26.1916017 
26.2106543 
26.2297541 
26.2433095 

26.2678511 

26.256.3739 
26.30.53929 
26.3245932 
26.3435797 
26.3623527 
26.331S119 
26.4tM375r6 
26.4196396 
26.4356031 

26.4575131 

26.4764046 

26.49.52526 

26.5141472 

26.5329933 

26.551,5361 

26.571  6605 

26..5594716 

26.60-2694 

26.6270539 

26.64-552.52 
26.6645333 
26.6533231 
26.7020593 
26.7207784 
26.73945.39 
26.7551763 
26.7765557 
26.7955220 
26.81417.54 

26.8.323157 
26.85144.32 
26.8700577 
26.  ,8536593 
26.9072481 
26.9255240 
26.9443572 
26.9629375 
26.9514751 
27.0000000 

27.0185122 
27.0370117 
27.0.5.549-5 
27.0739727 
27.0924344 
27.1105334 
27.1293199 
27.1477439 
27.1661554 
27.184.5544 

27.2029410 
27.22131.52 
27.2.396769 
27.2550263 
27.2763634 


8.8065722 
8.8108631 
8.8151598 
8.819^1474 
8.8237307 
8.8250099 
8.8322550 

8.8365559 
8.84(15227 
8.3450554 
8.8493440 
3.85359.35 
8.8575489 
8.8620952 
8.8663375 
8.8705757 
8.8748099 

8.8790400 
8.6532661 
8.8874582 
8.3917063 
8.8959204 
8.9001304 
8.9043:^6 
8.9035337 
8.9127369 
8.9169311 

8.9211214 
8.925.3073 
8.9294902 
8.9336687 
8.9375433 
8.9420140 
8.9461509 
8.9503433 
8.9545029 
8.9556581 

8.9623095 
8.9669570 
8.9711007 
8.9752406 
8.9793766 
8.9335089 
8.9876373 
8.9917620 
8.9953329 
9.0000000 

9.0041134 
9.0052229 
9.0123233 
9.0164309 
9.0205293 
9.0246239 
9.0287149 
9.0325021 
9.0365357 
9.0409655 

9.04.50419 
9.0491142 
9.0531831 
9.0572482 
9  0613098 


.ft01464129 
.001461933 
.001459854 
.001457726 
.001455604 
.001453483 
.001451379 

.001449275 
.001447173 
.001445087 
.00144.3001 
.001440922 
.00143-.549 
.001436782 
.0014:34720 
.0014:32665 
.0014.3C615 

.001423571 
.001426534 
.001424501 
.001422475 
.001420455 
.001418440 
.001416431 
.001414427 
.001412429 
.001410437 

.001403451 
.001406470 
.001404494 
.001402525 
.001400560 
.00139860! 
.001396648 
.001:394700 
001392758 
.001390821 

.001388889 
.001356963 
.00135.5042 
.001333126 
.001331215 
.001379310 
.001.377410 
.00137.5516 
.001373626 
.001371742 

.001369363 
.001:167959 
.001366120 
.001364256 
.roi 362398 
.001360544 
.001355696 
.001356352 
.001355014 
.001353180 

.001351351 
.001349528 
.001347709 
.001:345.-95 
.001344036 


15U 

TABLE  XI    SQUARE 

S,  CUBES, 

SQUARE  R( 

)OTS, 

No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocala. 

745 

555025 

4134936i5 

27.2946331 

9.0653677 

.001342232 

746 

556516 

415160936 

27.3130006 

9.0694220 

.001340433  . 

747 

55S039 

416332723 

27.3313007 

9.0731726 

.001333638 

74S 

559504 

413503992 

27.3495337 

9.0775197 

.001336-93 

749 

561031 

4201 39749 

27.3673644 

9.0315631 

.001335113 

750 

562500 

421375000 

27.3361279 

9.0356030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.0396.392 

.001331553 

752 

565501 

425259003 

27.4226134 

9.0936719 

.001329737 

753 

567009 

426957777 

27.4403455 

9.0977010 

.001323021 

754 

563516 

423661064 

27.4590604 

9.1017265 

.001326260 

755 

570025 

430363375 

27.4772633 

9.1057435 

.001324503 

756 

571536 

432031216 

27.4954542 

9.1097669 

.001322751 

757 

573  )49 

433793093 

27.5136330 

9.1137818 

.001321004 

75S 

574564 

435519512 

27.5317993 

9.1177931 

.031319261 

759 

576031 

437245479 

27.5499546 

9.1213010 

.001317523 

7^,0 

577600 

433976000 

27.5630975 

9.1253053 

.001315739 

761 

579121 

440711031 

27.5362234 

9.1293061 

.001314060 

76-2 

5S0644 

442450723 

27.6343475 

9.13.330a4 

.001312336 

763 

532169 

444194947 

27.6224546 

9.1377971 

.001310616 

764 

583696 

445943744 

27.6405499 

9.1417374 

.001303901 

765 

535225 

447697125 

27.6536334 

9. 1457742 

.001307190 

766 

536756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

533239 

451217663 

27.6947643 

9.1537375 

.001-303781 

763 

539324 

452934332 

27.7123129 

9.1.5771.39 

.001302033 

769 

591361 

454756609 

27.7303492 

9.1616369 

.001300390 

770 

592900 

456533000 

27.74337.39 

9.16.56565 

.001293701 

771 

594441 

453314011 

27.7663363 

9.1696225 

.001297017 

772 

595934 

463399648 

27.7843330 

9.17353.52 

.001295337 

773 

597529 

461839917 

27.3023775 

9.1775445 

.00129:^661 

774 

599 i-e 

463634324 

27.8203555 

9.1315003 

.001291990 

775 

603625 

465434375 

27.8333218 

9.18.54527 

.001290323 

776 

602176 

467233576 

27.3567766 

9.1394018 

.001233660 

777 

603729 

469397433 

27.8747197 

9.1933474 

.001237001 

773 

605234 

470910952 

27.3926514 

9.1972397 

.00123.5.347 

779 

636341 

472729139 

27.9105715 

9.2012236 

.001233697 

780 

633400 

474552000 

27.9234301 

9.2051641 

.001282051 

73 1 

609961 

476379541 

27.9463772 

9.2090962 

.001230410 

732 

611521 

473211763 

27.'J042629 

9.21302.50 

.031278772 

783 

613039 

430043637 

27.9321372 

9.2169505 

.001277139 

784 

614656 

431590304 

28.00313030 

9.2203726 

.001275510 

733 

616225 

433736625 

23.0178515 

9.2247914 

.001273385 

736 

617796 

435537656 

23.0.356915 

9.2237063 

.001272265 

737 

619369 

43744:J403 

23.0535203 

9.2.326189 

.001270648 

733 

620944 

4393)3372 

23.0713377 

9.236.5277 

.001269036 

789 

622521 

491169069 

23.03914.33 

9.2404333 

.001267427 

790 

624100 

493039000 

23.1069336 

9.244.3355 

.001265323 

791 

625631 

494913671 

23.1247222 

9.2432344 

.001264223 

792 

627264 

496793033 

28.1424946 

9.2.521300 

.001262626 

793 

623S49 

493677257 

23.1602.557 

9.2560224 

.0012610:J4 

794 

630436 

503566134 

23.17303-56 

9.2599114 

.0312.59446 

795 

632325 

502459375 

23.1957444 

9.2637973 

.001257362 

796 

633616 

504353336 

23.2134720 

9.2676793 

.001256281 

797 

635209 

506261573 

28.2-311834 

9.2715592 

.001254705 

793 

636304 

503169592 

23.2433933 

9.2754.3.52 

.0012.5313? 

799 

633401 

510032399 

23.266.5831 

9.2793031 

.001251564 

300 

640000 

512000000 

28.2342712 

9.2831777 

.001250000 

801 

641601 

513922431 

23.3019434 

9.2370440 

.0012434.39 

802 

643204 

515349603 

23.3196045 

9.2909072 

.001246333 

803 

644309 

517731627 

23.3372546 

9.2947671 

.0012453.30 

sai 

646416 

519713464 

23.3.543933 

9.2936239 

.001243781 

805 

643025 

521660125 

23.372.5219 

9.3024775 

.001242236 

806 

649636 

523636616 

23.3901391 

1   9.3063273 

.001240695 

CUBE    ROOTS,    AND    RECIPROCALS. 


151 


No. 

Squares. 

Cubes.    i 

Square  Hoots. 

Cube  Roots. 

Reciprocals. 

807 

651219 

.525557943 

23.4077454 

9.3101750 

.001239157 

803 

652364 

527514112 

28.4253403 

9.3140190 

.001237624 

809 

6.54431 

529475129 

23.44292.33 

9.3178599 

.001236094 

810 

656100 

531441000 

23.4604939 

9.3216975 

.001234563 

811 

657721 

533411731 

23.4780617 

9.325.5320 

.001233046 

812 

6593 14 

5353S7323 

23.4956137 

9.3293634 

.001231527 

813 

660369 

537367797 

23.5131549 

9.3331916 

.001230012 

814 

662596 

.539353144 

23.5.3063.52 

9..3370167 

.001223.301 

815 

661225 

541343375 

23.. 5432043 

9.3403336 

.001226994 

816 

665356 

543338496 

23.5657137 

9.3146575 

.00122.3490 

817 

6674S9 

545333513 

23.  .533211 9 

9.3434731 

.001223990 

81^ 

66 J 124 

547343432 

23.6006993 

9.3522357 

.001222494 

819 

670761 

549353259 

23  6131760 

9.3560952 

.001221001 

820 

67240) 

5513630X 

23.63.56421 

9.3599016 

.001219512 

821 

674041 

553337661 

23.6530976 

9.3637049 

.001213027 

822 

675634 

555412213 

23.6705424 

9.3675051 

.001216545 

823 

677329 

557441767 

23.6S79766 

9.3713022 

.00121.3067 

824 

673976 

559476224 

23.7054002 

9.3750963 

.001213.592 

825 

6S0625 

561515625 

23.7223132 

9.3733373 

.001212121 

826 

632276 

563559976 

23.7402157 

9.33267.32 

.001210654 

827 

6S3929 

565609233 

23.7576077 

9.3364600 

.001209190 

82S 

635534 

567663552 

23.7749391 

9.3902419 

.001207729 

829 

6S7241 

569722739 

23.7923601 

9.3940206 

.001206273 

830 

633900 

571737000 

23.8097206 

9..3977964 

.001204319 

831 

690561 

573356191 

23.8270706 

9.4015691 

.001203369 

832 

692224 

575930363 

23.3444102 

9.40533S7 

.001201923 

833 

693339 

578009537 

23.3617394 

9.4091054 

.001200430 

834 

695556 

580093704 

23.3790532 

9.4123690 

.001199041 

835 

697225 

532132375 

23.S963666 

9.4166297 

.001197605 

836 

693396 

534277036 

23.9136646 

9.4203373 

.001196172 

837 

700569 

536376253 

23.9309523 

9.4241420 

.001194743 

S3S 

702244 

533430472 

23.9432297 

9.4278936 

.001193317 

839 

703921 

590:89719 

23.96.54967 

9.4316423 

.001191395 

840 

705600 

592704000 

23.93275.35 

9.4353380 

.001190476 

841 

7072SI 

594323321 

29.0000000 

9.4.391.307 

.001139061 

842 

703964 

596947633 

29.0172.363 

9.4123704 

.001187643 

843 

710349 

599077107 

29.0344623 

9.4466072 

.001136240 

844 

712336 

601211534 

29.0516731 

9.450.3410 

.001184334 

845 

714025 

603331125 

29.0633337 

9.4510719 

.001183432 

846 

715716 

605495736 

29.0360791 

9.4577999 

.001132033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001130633 

848 

719104 

609300192 

29.1204396 

9.46.52470 

.001179245 

849 

720301 

611960049 

29.1376046 

9.4639661 

.001177856 

850 

722500 

614125000 

29.1.547595 

9.4726324 

.001176471 

831 

724201 

616295051 

29.1719043 

9.4763957 

.001175033 

852 

725904 

613470203 

29.1390390 

9.4301061 

.001173709 

853 

727609 

620650477 

29.2061637 

9.4333136 

.001172333 

854 

729316 

622335364 

29.2232734 

9.4375182 

.001170960 

855 

731025 

625026375 

29.2403331 

9.4912200 

.001169.591 

856 

732736 

627222016 

29.2574777 

9.4949133 

.001163224 

857 

734449 

629122793 

29.274.5623 

9.4936147 

.001166361 

853 

736164 

•   63162>712 

29.2916370 

9.5023073 

.001163.501 

859 

737331 

633339779 

29.3037018 

9.50.59930 

.001164144 

860 

739600 

636056000 

29.3257566 

9.5096354 

.001162791 

861 

741321 

633277331 

29.3423015 

9.51.33699 

.001161440 

862 

743044 

610503923 

29.3593365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3763616 

9.5207303 

.001153749 

864 

746 196 

6 14972544 

29.3933769 

9.5244063 

.001157407 

865 

743225 

617214625 

29.4103323 

9.  .5230794 

.001156069 

866 

749956 

619161896 

29.4273779 

9.5317197 

.001154734 

867 

751639 

651714363 

29.4443637 

9.53.34172 

.001153403 

863 

, 

753424 

'   653972032 

29.4613397 

9.5390318 

.001152074 

152 


TABLE  XI. 


SQUARES,  CUBES,  SQUARE  KOO/S, 


No. 


S69 

870 
871 
872 
873 
874 
875 
876 
877 
878 
879 

880 
88 1 
882 
8S3 
8.S4 
SSo 
856 
837 
8.38 
SS9 

890 
891 
892 
893 
894 
895 
896 
897 
893 
899 

900 
901 
902 
903 

9m 

9135 
906 
907 
90S 
909 


Squares. 


920 
921 
922 
923 
924 
925 
926 
927 
923 
929 
930 


755161 

756900 
7;:?&41 
7603S4 
762129 
76;JS76 
765625 
767376 
769129 
770S34 
772641 

774400 
776161 
777924 
7796S9 
781456 
7S3225 
784996 
7S6769 
78S.544 
790321 

792100 
793>S1 
795664 
797449 
799236 
801f!25 
802S16 
804609 
806404 
80S20I 

810000 
81IS01 
813604 
815409 
817216 
819025 
820S36 
822649 
824464 
826231 


Cubes. 


910 

S2S10V-) 

911 

829921 

912 

S3 1 744 

913 

83:3569 

914 

835396 

915 

837225 

916 

839056 

917 

&103S9 

918 

842724 

919 

844561 

S46400 
84S241 
850054 
851929 
853776 
855625 
857476 
859329 
861154 
863041 
864900 


Square  Roots. 


656234909 

65.S503000 
660776311 
6630.54S43 
665335617 
667627624 
669921575 
672221376 
674526133 
676-36152 
679151439 

651472000 
6S3797S41 
6-612S965 
65S4653S7 
69OS071O4 
693154125 
695506456 
697864103 
7ai227072 
702595369 

704969000 
707347971 
709732258 
712121957 
714516954 
716917375 
719:323136 
7217:34273 
724150792 
726572699 

729000000 
731432701 
73-35705(:«3 
7:36314:327 
7-3576:3264 
741217625 
74:3677416 
746142643 
74561:3312 
751059429 

753571000 
756055031 
75555052S 
761045497 
76:3551944 
766360575 
76-575296 
771095213 
77:362(;'632 
776151559 

77S65.5000 
781229961 
7.53777445 
756:330467 
785559024 
791453125 
794022776 
796597S53 
79917S752 
801765059 
804357000 


Cube  Roots. 


29.4788059 

29.4957624 
29.5127091 
29.5296461 
29..S1657.^ 
29.56:34910 
29.580:3959 
29.5972972 
29.6141555 
29.6310&JS 
29.rA79M2 

29.6647939 
29.6516442 
29.6934545 
29.7153159 
29.7:321:375 
29.7459496 
29.7657521 
29.732.54.52 
29.799:3259 
29.3161030 

29.5:325678 
29.5496231 
29.566:3690 
29.5531056 
29.3995.328 
29.916-5506 
29.93:32-591 
29.9499-533 
29-9666431 
29.95:3.3257 

30.0000000 
30.0166620 
30.03:33143 
30.0499554 
3(».  0665923 
30.03:32179 
30.099-:339 
30.1164407 
30.1:3:30:353 
30.1496269 

30. 1662063 
30.1527765 
30-199:3-377 
30.21-55599 
30.2-324:329 
30.2459669 
3<t.  26.549 19 
30.2320079 
30.29-5143 
30.31-50123 

30-3-31-5013 
30.^479313 
30.3644529 
30..3309151 
30.397:3653 
30.41-35127 
30.4302451 
30.4466747 
30.46-30924 
30.4795013 
30.4959014 


Reciprocals. 


9.5427437 

9.5464027 
9.5500539 
9.5-537123 
9.55736:30 
9.5610103 
9. -5646.559 
9.5652932 
9.5719377 
9.5755745 
9.5792055 

9..552>:397 
9.5564632 
9.5900939 
9.5937169 
9.-5973373 
9.6009.545 
9.604-5696 
9.6051517 
9.6117911 
9.615:3977 

9.619f)017 
9.6226030 
9.6262016 
9.6297975 
9.6:3:3:3907 
9.6369512 
9.64(t5690 
9.&44h542 
9.6477:367 
9.65131C6 

9.6.5459.33 

9.6-534654 

9-6620403 

9.66-5G096 

9.6691762  ' 

9.6727403 

9.6763017 

9.6795604 

9.6534166 

9.6.569701 

9.690521 1 
9.6940694 
9.6976151 
9.701 1;583 
9.7046959 
9.7052-369 
9.7117723 
9.71^3051 
9.713-:354 
9.722:3631 

9.7255853 
9.7294109 
9.7329:309 
9.7364454 
9.7399634 
9.74:^753 
9.7469557 
9.7.504930 
9.7539979 
9.7575002 
9.7610001 


.001 1.50743 

.001149425 
.001145106 
.001146759 
.001145475 
.001144165 
.001 142357 
.001141553 
.001140251 
.001 13-952 
.0011:37656 

.00113n:i64 
.0011.3.5074 
.0011:3.3757 
.001132503 
.001131222 
.001129944 
.001123663 
.001127396 
.001126126 
.001124559 

.00112.3596 
.■001122:3:34 
-001121076 
.001119521 
.001113568 
.001117313 
.001116071 
.001114527 
.00111.3.556 
.001112:347 

.001111111 
.001109578 
.001105&47 
.001107420 
.0(01106195 
.001104972 
.00110.3753 
.001102536 
.001101.322 
.001100110 

.001095901 
.001097695 
.001096491 
.001095290 
.001094092 
.001092396 
.001091703 
.001090513 
.001059:325 
.001055139 

.001056957 
.001055776 
.001084.599 
.0010S:M23 
.001052251 
.0010810.31 
.001079914 
.001078749 
.001077556 
.001076426 
.00107.5269 


CUBE    ROOTS,     iND    RECIPROCALS. 


153 


No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

931 

866761 

806954491 

30.5122926 

9.7644974 

.001074114 

932 

S6S624 

809557563 

30.  .5236750 

9.7679922 

.001072961 

933 

870439 

812166237 

30.5450437 

9.7714345 

.001071811 

934 

872356 

814730501 

30.5614136 

9.7749743 

.001070664 

935 

874225 

817400375 

30.5777607 

9.7734616 

.001069519 

933 

876 J36 

820025356 

30..5941171 

9.7820466 

.001063376 

937 

877969 

822656953 

30.6104.557 

9.7854233 

.001067236 

93S 

879344 

825293672 

30.6267357 

9.7339037 

.001066098 

939 

831721 

827936019 

30.6431069 

9.7923361 

.001064963 

940 

833600 

833534000 

30.6594194 

9.7953611 

.001063330 

941 

835481 

833237621 

30.6757233 

9.7993336 

.001062699 

942 

8^7364 

835396333 

30.6920185 

9.30230.36 

.031051571 

943 

8^9249 

833561307 

30.7033051 

9.8062711 

.001060445 

944 

891136 

841232334 

3' (.7245330 

9.8097362 

.0010.59322 

945 

893')25 

843903625 

30.7403523 

9.8131989 

.001053201 

.  946 

894916 

S46590536 

30.7571130 

9.8166591 

.001057082 

947 

896309 

849273123 

30.7733651 

9.8201169 

.001055966 

943 

893704 

85197 L392 
854670349 

30.7896036 

9.8235723 

.0010.54852 

949 

9)0601 

30.8053436 

9.8270252 

.0010.53741 

950 

902500 

857375000 

33.3223700 

9.8.304757 

.0010.32632 

951 

901401 

860035351 

30.3332379 

9.3339233 

.001051525 

952 

9063  )4 

862301403 

30.8544972 

9.8373895 

.0010.50420 

953 

903209 

865523177 

30.8706931 

9.3403127 

.001049313 

954 

910116 

863250664 

.30.3863904 

9.8442536 

.001048213 

955 

912025 

870933375 

.30.9030743 

9.8476920 

.001047120 

956 

913936 

S73722316 

30. 9  1'j24  97 

9.8511230 

.031046025 

957 

915349 

876467493 

30.93.54166 

9.8545617 

.001044932 

958 

917764 

879217912 

30.9515751 

9.8579929 

.00104.3341 

959 

919631 

831974079 

30.9677251 

9.8614218 

.031042753 

960 

921600 

834736000 

30.9333663 

9.8643433 

.001041667 

961 

923521 

837503631 

31.0000000 

9.8632724 

.001043533 

962 

925444 

890277123 

31.0161243 

9.8716941 

.0010.39501 

963 

927369 

89305G347 

31.0322413 

9.8751135 

.0010.33422 

964 

929296 

895341344 

31.0433494 

9.8785305 

.001037344 

965 

931225 

893632125 

31.0644491 

9.8319451 

.001036269 

966 

933156 

901423696 

31.0835405 

9.8353574 

.001035197 

9o7 

935039 

904231063 

31.0966236 

9.8337673 

.001034126 

96^ 

937024 

907039232 

31.1126934 

9.8921749 

.001033353 

969 

933961 

909353209 

31.1237643 

9.89.55301 

.001031992 

970 

940301 

912673000 

31.1443230 

9.8939330 

.001030928 

971 

942-^11 

915493611 

31.1633729 

9.9023S35 

.001029366 

972 

9447S4 

913333043 

31.1769145 

9.9057317 

.001023307 

973 

946729 

921167317 

31.1929479 

9.9091776 

.001027749 

974 

943676 

924010424 

31.2039731 

9.912-3712 

.001026594 

j   97c 

950625 

926359375 

31.2249900 

9.91.59624 

.001023641 

J 

976 

952576 

929714176 

31.2409937 

9.9193513 

.001024590 

977 

954529 

932574333 

31.2560992 

9.9227379 

.001023.341 

973 

956434 

935441352 

31.2729915 

9.9261222 

.001022495 

979 

953441 

93S313739 

31.2339757 

9.9295042 

.001021450 

930 

960400 

941192003 

31. .304951 7 

9.9323339 

.001020403 

931 

962361 

944076141 

31.3209195 

9.9362613 

.031019363 

932 

964324 

946066! 63 

31.3363792 

9.9396.363 

.001018330 

933 

9662S9 

949;:62337 

31.3523303 

9.94.30092 

.001017294 

94 

96325'6 

952763904 

31.3637743 

9.9463797 

.001016260 

935 

970225 

955671625 

31.3347097 

9.9497479 

.001015223 

'■j-6 

972196 

9535^52.56 

31.4006369 

9.9.531133 

.031014199 

9S7 

974169 

951504303 

31.4165.361 

9.9.564775 

,001013171 

93S 

976144 

964430272 

31.4324673 

9.9593389 

.001012146 

939 

978121 

967361669 

31.4433704 

9.9631931 

.001011122 

990 

930100 

970299000 

31.4642654 

9.9665549 

.001010101 

991 

932031 

973242271 

31.430152:5 

9,9699095 

.001009082 

992 

934064 

976191433 

31.4960315 

9.9732619 

.001008065 

1D± 

TABLE    XI.       SQUARES,    CUBES,    &C. 



No. 

Squares. 

Cubes. 

Square  Roots. 

Cube  Roots. 

Reciprocals. 

993 

956049 

9791466.57 

31. .51 19025 

9.9766120 

.001007049 

994 

933036 

982107784 

31.5277655 

9.9799.599 

.001006036 

99.5 

990025 

935074375 

31.54.36206 

9.93.33055 

.001005025 

996 

992016 

933047936 

31.5594677 

9.9&66488 

.001004016 

997 

994)09 

991026973 

31.5753063 

9.9899900 

.001003009 

993 

996004 

994011992 

31.5911380 

9.9933289 

.001002004 

999 

993001 

997002999 

31.606.613 

9.9966656 

.001001001 

1000 

1000000 

1000000000 

31.6227766 

10.0000000 

.001000000 

1001 

1002001 

1003J0.3001 

31.6.33.5340 

10.0033322 

.0009990010 

1002 

1004004 

1006012003 

31.6.543836 

100066622 

.0009980040 

1003 

10i'6!09 

1009027027 

31.67017.52 

10 0099599 

.0009970090 

1       1004 

100S0I6 

1012043064 

3 1.63  59:' 90 

lOO  1.331 55 

.0009960159     1 

1005 

1010025 

101.5075125 

31.7017319 

100166339 

.0009950249     ! 

!       1006 

1012036 

1018103216 

31.7175030 

10.019C60] 

.0009940358     i 

1007 

1014049 

1021147.343 

31.7332633 

10.02.32791 

.0009930487     j 

1003 

1016064 

1024192512 

31.74901.57 

IO02659.53 

.0009920635 

1009 

1018031 

1027243729 

31.7647603 

100299104 

.0009910803 

1010 

1020100 

1030.301003 

31.7804972 

10.0.332228 

.0009900990 

j       1011 

1022121 

1033.364331 

31.7S^62262 

10.0.3653.30 

.0009391197 

i       1012 

1024144 

10.36433723 

31.8119474 

100393410 

.0009881423 

1013 

1026169 

1039.509197 

31.8276609 

1O0431469 

.0009371668 

1014 

1023196 

1W2590744 

31.84.3.3666 

100464.506 

.0009561933 

1015 

1030225 

104.5678375 

31.8590646 

10.0497521 

.0009852217 

1016 

1032256 

1048772096 

31.8747.549 

1O0.530514 

.0009-842520 

1017 

10342S9 

1051871913 

31.8904374 

10.0563435 

.0009832842 

1018 

1036324 

10.54977332 

31.9061123 

10.0596435 

.0009323133  ■ 

1019 

1033361 

1053039359 

31.9217794 

100629364 

.0009813543 

1020 

\yinioz 

r06 1203000 

31  9374388 

10.0662271 

.0009803922 

1021 

i'34244I 

1064332261 

31.9.530906 

1006951.56 

.0009794319 

1022 

1044434 

1067462643 

31.9637347 

10.072-020 

.0009784736 

1023 

1046529 

1070599167 

31.984.3712 

100760-63 

.0009775171 

1024 

1013576 

1073741324 

32.0000000 

10.0793634 

.0009765625 

1025 

1050625 

1076^90625 

32.0156212 

10.0326434 

.0009756098 

iOv6 

1052676 

1030045576 

32.0.3*2:343 

100-59262 

.0009746589 

1       1027 

10.54729 

1083206633 

32.0463107 

10.0392019 

.0009737098 

1023 

10.56784 

10S6373952 

32.0624391 

10.0924755 

.0009727626 

1029 

10.58341 

1039.547389 

32.0730293 

10.0957469 

.0009718173 

■       1030 

1060900 

1092727000 

32.09.36131 

10.0990163 

.000)9708738 

1031 

1062961 

109.5912791 

32.1091337 

10.10228.35 

.0009699321 

1032 

106.3024 

1099104763 

.32.1247563 

lO105r>187 

.0009639922 

1033 

1067039 

1102.3029.37 

32.1403173 

10 10381 17 

.0009680542 

1031 

1069156 

1105.507.304 

.32.1.5.58701 

10.1120726 

.0009671180 

1035 

1071225 

1108717375 

32.1714159 

1011.5.3314 

.0009661836 

1036 

1073296 

1111934656 

32.18695.39 

10.118.5832 

.0009652510 

I       1037 

1075.369 

11151.576.53 

32.2024344 

10.1218428 

.0009643202 

1033 

1077444 

11133>56872 

32.2180074 

10.12.509.53 

.00096.3391 1 

1039 

1079.521 

1121622319 

32.2335229 

101283457 

.0009624639 

1040 

I0316!J0 

1124364000 

32.2490310 

10.131.5941 

.000961.5335 

H"41 

1033631 

1123111921 

32.264.5316 

101.343403 

.0009606143 

li42 

1035761 

1131366033 

32.2800248 

1O1.3S0345 

.0009596929 

if  43 

1037349 

1134626507 

32.2955105 

lO  1413266 

.0009587738 

1044 

1039936 

1137393134 

32.3109338 

10.1445667 

.0009578.514 

1045 

1092125 

1141166125 

32.3264598 

101478047 

.0009569378 

1046 

1094116 

114444.5.336 

32.3419233 

101510406 

.0009560229 

1047 

1096209 

1 147730-23 

32.3573794 

101.542744 

.0009551098 

104S 

1093.304 

1151022592 

32.3723231 

10.1575062 

.0009541985 

1049 

1100401 

1154320649 

32.3882695 

10.1607.3.39 

.0009532888 

in50 

1102.500 

11.5762.50ao 

32.4037035 

lO  1639636 

.0069.52.3810 

1051 

1104601 

1160935651 

32.4191.301 

101671393 

.0009514748 

1052 

1106704 

1164252608 

32.434.5495 

10 1704 1 29 

.0009505703 

1053 

1103309 

1167575377 

32.4499615 

101736:M4 

.00O94S6676     | 

io.:4 

1110916 

117090.5464 

32.4653662 

10176^539 

.00OP437666 

f^.^ 

0  ./  0 

?    ! 

^j  ^,  t.i^  V  y   bC 

\ 

? 

i 

- 

1- 

A  ^                TABLE    XII. 

,/.  ^^^,.  .. 

■* 

" 

LOGARITHMS    OF    NUMBERi 

—  c;  // 

-* 

FROM   1  TO   10,000 

-.. 

^ 

4 

1 

\ 

156 

TABLE  XII.   LOGARITHMS 

Of 

NUMBERS. 

Ino.i 

0  1  1  1 
OOUOijG  000434] 

3 
000S63 

3 

001301 

4. 

5 

6  1  7  i  8 

9  iDiff. 

100 

001734 

002166 

002598003029  003461 

003691 

432 

1 

4321 

4751 

5181 

5609 

603- 

6466   6394! 

7321 1  7748 

8174|  428 

2 

8600 

90261 

9451 

9376  0103001 

010724  011147 

011570  011993 

0124151  424 

3' 

012S37 

013259 

0136S0 

014100: 

4521 

4940 

5360 

5779' 

6197 

6016 

420 

4 

7033 

74511 

7868 

82^41 

8700 

9110 

9532 

9947 

020361 

020775 

416 

5 

021189 

021603 

022016 

022423  022341 

023252 

023664 

024075 

4466 

4896 

412 

6 

5306 

5715; 

6125 

6533;  6942 

7350 

7757i 

8164 

8571 

8978 

408 

7 

93S4 

9789 

030195 

0.30600^ 

031(104 

031408 

031812' 

032216  032619! 

03:3021 

404 

8 

033424 
7426 

03:3326 

4227 

4623 

5029 

5430 

.5830 

6230 

6629 

702ft 

400 

9 

7825 

8223 

8620 ; 

9017 

9414 

9811 

040207 

040602 

040998 

397 

no 

041393 

041787 

042182 

1 

042576 

042969 

043362 

043755 

044148 

044540 

044932 

393 

1 

5323 

5714 

6105 

6195' 

63S5 

7275 

7664 

8053 

8442 

6830 

390 

2 

9218 

9606 

9993 

050380 

050766 

051153 

051538 

051924 

052309 

052694 

336 

3 

053076 

053463 

053^:46   4230 

4613 

4996 

5378 

5760 

6142 

6524 

383 

4 

6905 

72-6 

7606   80^6 

8426 

8805 

9185 

9563 

9942 

060320 

379 

5 

06069S 

061075 

061452  001829 

002206 

062582 

062958 

063333 

063709 

4083 

376 

6 

445S 

4S32 

5206 

55S0 

5953 

6326 

6099 

7071 

7443 

7815 

373 

7 

8186 

8557 

892S 

9293 

9663 

070038 

070407 

070776 

071145 

071514 

370 

8 

0718S2 

072250 

072617 

072985 

073352 

3718 

4085 

4451 

4816 

5182 

3G0 

9 

5547 

5912 

6276 

6640 

7004 

7303 

7731 

8094 

8457 

8819 

363 

120 

079181 

079543 

079904 

0S0266 

030626 

080987 

081347 

081707 

082067 

082426 

360 

1 

0327S5 

083144 

0S3503 

3361 

4219 

4576 

4934 

5291 

5647 

6004 

357 

2 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

3 

9905 

09025S 

090611 

090963 

0913t5 

091667 

092018 

092370 

092721 

093f!71 

352 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

6866 

6215 

6562 

349 

.  5 

6910 

7257 

7604 

7951 

8293 

86^14 

8990 

9335 

9631 

100026 

340 

6 

100371 

100715 

1010.59 

101403 

101747 

102091 

102434 

102777 

103119 

3462 

;343 

7 

3S04 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6371 

341 

8 

7210 

7549 

7883 

8227 

8565 

8903 

9241 

9579 

9910 

n0253 

338 

9 

110590 

110926 

111263 

111599 

1119:34 

112270 

112605 

112940 

113275 

3609 

335 

130 

1139t3 

114277 

il4611 

114944 

115278 

11.5611 

11.5943 

116276 

11600ft 

116940 

333 

1 

7271 

7603 

7931 

8265 

8595 

8926   9256! 

9586 

9915 

120245 

330 

2 

120574 

120903 

121231 

121560 

121838:12221'>|122544| 

122871 

123198 

3525 

328 

3 

3S52 

4178 

4504 

4330 

5156 

.5481   5806 

6131 

6-356 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722   9045 

9368 

9690 

130012 

323 

5 

130334 

130655 

130977 

131298 

131619 

131939  132260 

132580 

132900 

3219 

321 

6 

3539 

3-^5S 

4177 

41P6 

4314 

5133   5451 

5769 

60S6 

6403 

318 

7 

6721 

7037 

7354   7671 

7987 

8.303   8r,18 

8934 

9249 

9564 

316 

8 

9S79 

140194 

14050S 

140822 

141136 

141450  141763 

142076 

142369 

14270:<; 

314 

9 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

311 

140 

14612S 

14643S 

146743 

147058 

147.367 

147676 

147985 

148294 

148603 

148911 

309 

1 

9219 

9527 

9835 

150142 

150149 

150756 

151063 

151:370 

151676 

151952 

307 

2 

1522SS 

152594 

1529(10 

3205 

3510 

.3315 

4120 

4424 

4728 

5032 

305 

3 

5336 

.56411 

5943 

,  6246 

6549 

6352 

7154 

7457 

7759 

6061 

303 

4 

8362 

86f54 

8965 

9266 

9567 

936-^ 

160163 

100469 

160769 

16106S 

301 

5 

161 36S 

161667 

161967 

1162266 

162564 

162363 

3161 

3460 

37.58 

4055 

299 

6 

4353 

4650 

4947 

:  5244 

5.541 

5833 

61:34 

6430 

6726 

7022 

297 

7 

7317 

7613 

790s 

i  8203 

8497 

8792 

9086 

9360 

9674 

9968 

295 

8 

170262 

170555 

170848 

171141 

171434 

171726 

172019 

172311 

172603 

172895 

293 

9 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

150 

176091 

1763^1 

176670 

176959 

177248 

177536 

r7325 

178113 

178401 

178689 

289 

1 

8977 

926 1 

9552   9839 

,180126 

180413 

180699 

180986 

181272 

181558 

287 

2 

181 844 

182129 

182415  18270!) 

2985 

3270 

3555 

3f339 

4123 

4407 

285 

3 

4691 

4975 

.5259   5.542 

5S25 

6108 

6.391 

6674 

6956 

7239 

283 

4 

7521 

7S03 

8n84   8366 

8R47 

8928 

9209 

9490 

9771 

190051 

231 

5 

190332!  19061 2 

190-^92  191171 

191451 

1917.30 

192010 

192289 

192567 

2S46 

279 

6 

3125   3403 

3681   3959 

4237 

4514 

4792 

5069 

.5346 

5623 

278 

7 

5900   6176 

6453   6729 

7005 

7231 

7556 

7832 

8107 

8382 

276 

8 

8657'  8932 

9206   94SI 

9755 

200029 

200303 

200577 

200850 

201124 

274 

9 

{NO. 

201397 
0 

201670 

201943 

20*22 16 
I  3 

;  2024 38 

2761 

3033 

3305 

3577 

3848 

272 
Diff. 

1 

a 

I  * 

5 

6 

7 

8 

9 

TABLE    XII.       LOGAPJTHMS    OF    NUMBERS. 


[j1 


No.!     O 

1 

2 
3 


GS26 
9515 

2I21S3 

7434 
22a 1 OS 
2716 
5309 

7SS7 


3 


170 

11 

2; 

3 
4! 
5 
6 
7 
S 
9 


230149 

299G 
5o-23 
8016 

210ol9 
30:}S 
5513 
7073 

25012  » 
2353 


204391 
7096' 
9783 

212154 
5109 
7747 

220370' 
2976| 
5563 
S144 

230701 
3250 


130, 

i! 

2 
3 

4 

5' 

S\ 

7 

8 

9 


255273 
7679 

260:J7l 
2451 
4313 
7172 
9513 

271312 
4153 
6162 


5731 
3297 

210799 
3236 
5759 
R219 

250661 
3J96 


2.)1663 
7365 

210)51 
2720, 
5373 1 

soio; 

220631: 
32361 
5326' 
8409 

230959 
3501 
6033 

8543 

241043 

3531 

6096 

8461 

250903 

3333 


2019311 
7631 

210319 
2936 
5633 
8273 

220392 
3196 
60311 
8657 

231215 

3757 
6235 
8799 

241^297 
3782 
6252 
8709 

251151 
3530 


205204 
7901 

210536 
3252 
5902 
8536 

221153 
3755 
6312 
8913 


20547 
3173 

210353 
3513 
6166 
8793 

221414 
4015 

66o;) 

9170 


8 


205746 
84411 

211121' 
3783 
6130 
9060 

221675 
4274 
6858 
9426 


190  278754 
1231033 
2!  3301 


7802 
290035 
2256 
4466 
6665 
8353 


255514 
7913 

260310 
2633 
5051 
7406 
9746 

272074 
4339 
6692 

278932 
231261 
3527 
5782 
8026 
290257 
2473 
4637 
6334 
9071 


2J0/0-J 

8153 
260513 
2925 
5290 
7611 
9930 
272303 
4620 
6921 


255996 

8393 
260787 
3162 
5525 
7875 
'270213 
2533 
4350 
7151 


231470 
4011 
6537 
9049 

211516 
4030 
6199 
8951 

25139 
3322 

256237 

8637 
261025 
3399 
5761 
8110 
270116 
2770 
5031 
7330 


^00  301030 


3196 
5351 
7496 
9639 
311751 
3367 
5970 
8)63 


9,320146 


210 


322219 


30124 
3112 
5566 
7710 
9343 

311966 
4073 
6130 
8272 

320354 

322426 


1 

4232 

4433 

2 

6336 

6541 

3 

8330 

8533 

4 

330114 

330617 

5 

2433 

2640 

6 

4451 

4655 

7 

6160 

665) 

8 

8456   8656 

9 

310144  310612 

No. 

0 

1 

27921 1 
231433 
3753 
6007 
8219 
290430 
2699! 
4907' 
7104 
9239 

301464 
3623 
5781 
7924 

310056 
2177 
4259 
6390 
8131 

32J562 

322633 

4691 
6745 
8787 

330319 
2312 
4356 
6360 
8355 

310311 

3 


231721 
4261 
6739 
9299 

241795 
4277 
6745 
9193 

251633 
4064 

256177 
8877 

261263 
3636 
5996 
8344 

270679 
3001 


279439 

231715 

3979 

6232 

8473 

290702 

;  2920 

I  5127 

7323 

9507 


279667 
231942 
4205 
6456 
8696 
290925 
3141 
5347 
7542 
9725 


5311 
.  7609 

279395 
232169 
4431 
6631 
8920 
291147 
3363 
5567 
7761 
9943 


231979 
4517 
7041 
9550 

242044 
4525 
6991 
9443 

251881 
4306 

256718 
9116 

261501 
3373 
6232 
8578 

270912 
3233 
554? 


206016 
8710 

211333 
4049 
6691 
9323 

221936 
4533 
7115 
9632 

232234 

4770 
7292 
9300 

242293 
4772 
7237 
9637 

252125 
4543 


9 


206236 
8979! 

211651 
4314 
6957 
9535 

222196 
4792 
7372 
9933 


Diff. 


206556 
9247 

211921 
4579 
7221 
9346 

222456 
5051 
7630 

230193 


301631  301893 
3344  4059 
599Gi  6211 
8 137 1  8351 

3102631310431 


256953 
9355 

261739 
4109 
6467 
8812 

271144 
.3464 
5772 
8067 


232488 
5023 
7541- 

240050 
2541 
5019 
7432 
9932 

252363 
4790 

257193 
9591 

261976 
4346 
6702 
9046 

271377 
3696 
6002 
8296 


230123 
2396 
4656 
6905 
9143 

291369 
3584 
5787 
7979 


232742 
5276 

7795 
240300 
2790 
5266 
77231 
250176 
2610 
5031 

257439 
9333 

262214 
4532 
6937 
9279 

271609 
3927 
6232 
8525 


2339 
4499 
6599 
8639 
320769 

322339 
4399 
6950 
8991 

.331022 
.3011 
5057 
7060 
9054 

311039 


2609 
4710 
6309 
8393 
320977 

323046 
5105 
7155 
9194 

331225 
3216 


302114 

4275 
6425 
8561 

310693 
2312 
4920 
7018 
9106 

321181 


300161 

302331 
4491 
6639 
8778 

310906 
3023 
5130 
7227 
9314 

321391 


230351 
2622 
4332 
71.30 
9366 

29159! 
3301 
6007 
8193 

300373 

302517 
4706 
6351 
899 

311113 
3231 
5340 
7436 
9522 

321593 


271 

269 
267 
266 
261 
262 
261 
259 
253 
256 

255 
253 
252 
250 
219 
248 
246 
245 
243 
242 

241 
239 
233 
237 
235 
234 
233 
2.32 
23C 
229 


5257 

7260 

9253 

311237 


323252 
5310 
7359 
9393 

.331427 
3447 
5458 
7459 
9451 

3414.35 


323453 
5516 
7563 
9601 

331630 
3619 
5653 
7659 
96.50 

341632 


230573 
2349 
5107 
7354 
9539 

291813 
4025 
6226 
8416 

300595 

302764 
4921 
7063 
9204 

311330 
3445 
5551 
7616 
973n 

32130." 


280806 
3075 
5332 
7578 
9812 

292031 
4246 
6446 
8635 

300313 

302930 


5136 

7232 
9417 

311542 
3656 
5760 
7854 
9933 

.322012 


323665 
5721 
7767 
9305 

331S32 
3350 
5859 
7853 
9349 

341330 


323371 
5926 
7972 

330003 
2034 
4051 
6059 
8053 

340047 
2023 

8 


324077 
6131 
8176 

3.30211 
2236 
4253 
6260 
8257 

340246 
2225 

9 


228 
227 
226 
225 
223 
222 
221 
220 
219 
218 

21i 

216 

215 

213 

212 

211 

210 

209 

203 

207 

206 
205 
204 
203 
202 
202 
201 
200 
199 

otff.ji 


158 


TABLE    XII.       LOGARITHMS    OF   NUMBERS. 


No. 

220 

0 

1 
342620 

3 
342317 

3  I 

4: 

343212 

5 

6 

7 
343302 

8 

343999 

9 

Diff. 

31^423 

343014 

343409 

343606 

344196 

197 

1 

4392 

4539 

4785 

4981 

6178 

5374 

5570 

5766 

5S62 

6157 

196 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9360 

350054 

194 

4 

350248 

350442 

350636 

350829 

351023 

351216 

351410 

351603 

351796 

1939 

193 

5 

2183 

2375 

2563 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

4103 

4301 

4493 

4635 

4876 

5063 

5260 

5452 

6643 

5834 

192 

7 

6026 

6217 

64G3 

6599 

6790 

6931 

7172 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506) 

8696 

8336 

9076 

9266 

9456 

9646 

190 

9 

9835 

360025 

360215 

360404 

360593 

360783 

360972 

361161 

361350 

361539 

1S9 

230 

361723 

361917 

362105 

362294 

362482 

362671 

362859 

363048 

363236 

363424 

188 

1 

3612 

3S00 

3933 

4176 

4363 

4551 

4739 

4926 

6113 

5301 

138 

2 

5483 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

95S7 

9772 

9953 

370143 

370328 

370513 

370698 

370383 

185 

5 

371063 

371253 

371437 

371622 

371806 

1991 

2175 

2360 

2544 

2728 

184 

6 

2912 

3096 

3230 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

133 

8 

6577 

6759 

6942 

7124 

7306 

7438 

7670 

7852 

8034 

8216 

132 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181 

240 

380211 

330392 

330573 

380754 

330934 

331115 

331296 

331476 

381656 

381837 

181 

1 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

ISO 

2 

3315 

3995 

4174 

4353 

4533 

4712 

4391 

5070 

5249 

5423 

179 

3 

5606 

5735 

5964 

6142 

6321 

6499 

6677 

6356 

7034 

7212 

178 

4 

7390 

7563 

7746 

7923 

8101 

8279 

8456 

S634 

8811 

8989 

178 

5 

9166 

9343 

9520 

9693 

9875 

390051 

390226 

390405 

390532 

390759 

177 

6 

390935 

391112 

391288 

391464 

391641 

1317 

1993 

2169 

2345 

2521 

176 

7 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176 

8 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175 

9 

6199 

6374 

6543 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

250 

397940 

398114 

398237 

398461 

393634 

398808 

393981 

399154 

399323 

399501 

173 

1 

9674 

9347 

400020 

400192 

400365 

400533 

40071 1 

400383 

401056 

401228 

173 

2 

401401 

401573 

1745 

1917 

2039 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3464 

3635 

3307 

3978 

4149 

4320 

4492 

4663 

171 

4 

4S34 

5005 

5176 

5346 

5517 

6688 

6858 

6029 

6199 

6370 

171 

5 

6540 

6710 

6381 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170 

6 

8240 

8410 

8579 

8749 

8918 

9037 

9257 

9426 

9595 

9764 

169 

7 

9933 

410102 

410271 

410440 

410609 

410777 

410946 

411114 

411283 

411451 

169 

8 

411620 

1738 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

163 

9 

3300 

3467 

3635 

3S03 

3970 

4137 

4305 

4472 

4639 

4806 

167 

260 

414973 

415140 

415307 

415474 

415641 

415808 

415974 

416141 

416308 

416474 

167 

1 

6641 

6307 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

166 

2 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165 

3 

9956 

420121 

420236 

420451 

420616 

420781 

420945 

421110 

421275 

421439 

165 

4 

421604 

1763 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

30S2 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4332 

5045 

5203 

5371 

5534 

5697 

5860 

6023 

61S6 

6349 

163 

7 

6511 

6674 

6336 

6999 

7161 

7324 

7486 

7648 

7311 

7973 

162 

8 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

9 

9752 

9914 

430075 

430236 

430398 

430559 

430720 

430881 

431042 

431203 

161 

270 

431364 

431525 

431635 

431846 

432007 

432167 

432328 

4324S8 

432649 

432809 

161 

1 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

160 

2 

4569 

4729 

4888 

50-18 

5207 

5367 

5526 

5635 

5844 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8067 

8226 

83S4 

8542 

8701 

8359 

9017 

9175 

153 

5 

9333 

9491 

9643 

9806 

9964 

440122 

440279 

440437 

440594 

440752 

158 

6 

440909 

441066 

441224 

441381 

441533 

1695 

1352 

2009 

2166 

2323 

157 

7 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

4513 

4669 

4825 

4981 

5137 

5293 

5449 

156 

9 
No. 

5694 

5760 

5915 

6071 
3 

6226 

6382 
5 

6537 
6 

6692 

6348 

7003 

155 

0 

1 

3 

4: 

7 

8 

9 

Diff. 

-^1 


TABLE    XII.       LOGARITHMS    OF    NUMBERS. 


159 


No   0 

1 
2 
3 

4 


290 
1 
2 
3 
4 
5 


447153 
87061 

450219 
17S6 
331S 
4S15 
6366 
78S2 
9392 

46JS'JS 

46239S 
3393 
53  S3 
6S63 
S347 
,  9322 
6  471292 


a 


2756 
4216 
5671 


447313 

8S61 
450403 
1940 
3171 
4997 
651S 
8033 
9543 
46104 

462543 
4042 
5532 
7016 
8495 
9969 

47143S 
2903 
4362 
5816 


447463 
90151 

450557 
2093 
3624 
5150 
6670 
8l8i 
9694 

461198 


447623 
9170 

450711 
2247 1 
3777| 
5302 
0S21 
8336 
«S45 

461348 


:447778 
1  9324 
1450365 
2100 
I     3930 


.300,477121 
1  8566 
2430007 


462697 
4191 
5630 
7164 
8643 

470116 
1535 
3019 
450S 
5962 


3 
4 
5 
6 

7 
8 
9 

310 
1 
2 
3 

4 
5 

6 

7 
8 
9 

320 

1| 

3 

^1 

6 

7 
8 
9 


462847 
4340 
5329 
7312 
8790 

470263 
1732 
3195 
4653 
6107 


5454 
6973 
6437 
9995 
461499 

462997 
4490 
5977 
7460 
8933 

470410 
187S 
3341 
4799 
6252 


8 


443242 
9787 

451326 
2859 
4387 
5910 
7428 
8940 

460447 
1943 


1413 
2S74 
4300 
5721 
7133 
8551 
9953 

491362 
2760 
4155 
5541 
6930 
831 
95S7 

501059 
2127 
3791 


477266 
871! 

480151 
15S6 
3016 
4142 
5363 
7230 
8692 


77411 


4 

«3oo 

450294 
1729 
3159 
4535 
6005 
7421 
8333 


463146 
4639 
6126 
7603 
9035 

470557 
2025 
3437 
4944 
6397 


490099  490239 


491502 
2900 
4294 
5633 
7063 
8443 
.  9324 
1501196; 
2564 
3927 


477555 
8999 

430433 
1872 
3302 
4727 
6147 
7583 
8974 

490330 


477700 
9143 

430582 
2016 
3445 
4369 
6289 
7704 
9114 

490520 


491612 
3040 
4433 
5322 
7206 
8586 
9962 

501333 
2700 
4063 


505150 
6505 
785i 
9203 
510515 
i  1833 
3218 
4513 
5874 
7196 


463296 
4783 
6274 
7756 
9233 

470704 
2171 
3633 
5090 
6542 


463445 
4936 
6423! 
7904 
9330 

470351 
2318 
3779 
5235 
6637 


448397 
9941 

4-31479 
3012 
4540 
6062 
7579 
9091 

460597 
2093 

463594 


448552 
450095 
1633 
3165 
4692 
6214 
7731 
9242 
460743 
2248 

463744 


491782 
3179] 
4572 
5960 
7344 
8724 

500099 
1470 
2837 
4199 


491922 


477844 
9287 

480725 
2159 
3587 
5011 
6430 
784; 
9255 

490661 

492062 


3319 
4711 
6099 
7483 
8362 
500236 1500374 
1744 


3453 

4350 
6233 
7621 
8999 


505236 
6640 
7991 
9337 

510679 
20171 
3351 
46S1 
6006 
7323 


330 
1 

2 
3 
4 
5 
6 

T 
I 

8 
9 


51S514 

9323 
521133 
2444 
3746 
5045 
6339 
7630 
3917 
V30200 


518646 
9959 

521269 
2575 
3376 
5174 
6469 
7759 
9045 


505421 
6776 
8126 
9471 

510313 
2151 
3434 
4313 
6139 
7460 

518777 

520090 

1400 

2705 

4006 


505557 
6911 
8260 
9606 

510947 
2234 
3617 
4946 
6271 
7592 


1607 
2973 
4335 

505693 
7046 
8395 
9740 

511031 
2418 
3750 
5079 
6403 
7724 


No.   O 


5304 
6593 
7888 
9174 
530456 

3 


520221 
1530 
2335 
4136 
5434 
6727 
8016 
9302 

530534 


519040 
520353 


3109 
4471 

505823 
7181 
8530 
9374 

511215 
2551 
3333 
5211 
6535 
7855 

519171 
520434 


477939 
9431 

430369 
2302 
3730 
5153 
6572 
7986 
9396 

490801 

192201 
3597 
4939 
6376 
7759 
9137 

590511 
1830 
3246 
4607 


478133 
9575! 

481012 
2445 
3872 
5295 
6714 
8127 
9537 

490941 


5035 
6571 
8052 
9527 
470993 
2464 
3925 
5331 
6332 

478278 
9719 

481156 
2588 
4015 


b3Jij 

8269 

9677 

491031 


5234 
6719 
8200 
9675 
471145 
2610 
4071 
5526 
6970 

478422 
9363 

481299 
2731 
4157 
5579 
6997 
8410 
9318 

491222 


492341 
3737 
5128 
6515 

7397 
9275 
500643 
2017 
3332 
4743 


492481 
3376 
5267 
6653 
8035 
9412 

5007S5 
2154 
3518 
4878 


505964 
7316 
8664 

510009 
1349 
2684 
4016 
5344 
6663 
7937 


492621 

4015 
540e 
6791 
8173 
9550 
500922 
2291 
3655 
5014 


u06099 
7451 
8799 

U10143 
1482 
2318 
4149 
5476 
6300 
8119 


Diff.  ' 

155 
154 
154 
153 
153 
152 
152 
151 
151 
150 

150 
149 
149 
148 
148 
147 
146 
146 
146 
14;' 

!45 
144 
144 
143 
143 
142 
142 
141 
141 
140 

140 
139 

139 

1391 

1381 

I33i 

1371 

137 

136 

136 


506234 
7536 
8934 

510277 
1616 
2951 
4282 
5609 
6932 
8251 


1661 

1792 

2966 

3096 

4266 

4396 

5563 

5693 

6356 

6935 

8145 

8274 

9130 

9559 

530712 

530340 

519303 
520615 
1922 
3226 
4526 
5822 
7114 
8402 
9637 
530963 

6 


520745 
2053 
3356 
4656 
.5951 
7243 
8531 
9815 

531096 


519566 
520376 
2183 
3486 
4785 
6031 
7372 
8660 
9943 
531223 

8 


506370 
7721 
9063 

510411 
1750 
3034 
4415 
5741 
7064 
8382 

519697 
521007 
2314 
3616 
4915 
6210 
7501 
8788 
530072 
135 


136 
135 
135 
134 
134 
133 
133 
133 
132 
132 

13! 

131 
131 
130 
130 
129 
129 
129 
128 
128 

Diff.i 


IbU 

TABLE  XII.   LOGARITHMS  OF 

.NUMBERS. 

No. 

340 

0 

531479 

1 
531607 

a 

3 

531862 

4: 

531990 

5 

6 

7 
532372 

8 

9 

Diff. 

123 

531734 

532117 

532245 

53250D 

532627 

1 

2754 

2332 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3S99 

127 

2 

4026 

4153 

4230 

4407 

4634 

4661 

4787 

4914 

5041 

5167 

127 

3 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

4 

6558 

6635 

6311 

6937 

7063 

7139 

7315 

7441 

7567 

7693 

126 

5 

7819 

7945 

8071 

8197 

8322 

8443 

8574 

8699 

8325 

8951 

126 

6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

540079 

540204 

125 

7 

540329 

540455 

540530 

r40705  540330 1 

540955 

541030 

541205 

1330 

1454 

125 

8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

9 

2S25 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124 

350 

544063 

544192 

544316 

544440 

544564 

544635 

544812 

544936 

545060 

545183 

124 

1 

5307 

5431 

5555 

5673 

5302 

5925 

6049 

6172 

6296 

6419 

124 

2 

6543 

6666 

67S9 

6913 

7030 

7159 

7282 

7405 

7529 

7652 

123 

3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

6881 

123 

4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9S61 

9934 

550106 

123 

5 

550223 

550351 

550473 

550595 

550717 

550840 

550S62 

551034 

551206 

1328 

122 

6 

1450 

1572 

1694 

1316 

1938 

2060 

2181 

2303 

2425 

2547 

122 

7 

2663 

2790' 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

r^i 

8 

3383 

4004 

4126 

4247 

4368 

4439 

4610 

4731 

4852 

4973 

121 

9 

5094 

5215 

5336 

5457 

5578 

6699 

5820 

5940 

6061 

6182 

121 

360 

556303 

556423 

556544 

556664 

5567S5 

556905 

557026 

557146 

557267 

5573S7 

120 

A 

7507 

7627 

774« 

7868 

7988 

8108 

8228 

8349 

8469 

8689 

120 

2 

8709 

8S29 

S94S 

9063 

9188 

9308 

9423 

9548 

S667 

9787 

120 

3 

9907 

560026 

560146 

560265 

560385 

560504 

560624 

560743 

560363 

560982 

119 

4 

561101 

1221 

1340 

1459 

1573 

1693 

1817 

1936 

2055 

2174 

119  j 

5 

2293 

2412 

2531 

2650 

2769 

2837 

3006 

3125 

3244 

3362 

119 

6 

3431 

3600 

3718 

3337 

3955 

4074 

4192 

4311 

4429 

4548 

119 

7 

4666 

4734 

4903 

5021 

5139 

5257 

5376 

5494 

6612 

5730 

lis 

8 

5348 

5966 

6034 

6202 

6320 

6437 

6555 

6673 

6791 

6S09 

118 

9 

7026 

7144 

7262 

7379 

.7497 

7614 

7732 

7849 

7967 

8084 

lis 

370 

563202 

568319 

563436 

568554 

563671 

568788 

568905 

569023 

569140 

569257 

117 

1 

9374 

9491 

960* 

9725 

9342 

9959 

570076 

570193 

570309 

570426 

117 

2 

570543 

570660 

570776 

570393 

571010 

571126 

1243 

1359 

1476 

1592 

117 

3 

1709 

1325 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116 

4 

2S72 

29.-;8 

3104 

3220 

3336 

3452 

3563 

3684 

3800 

3915 

116 

5 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

6 

5183 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

7 

6341 

6457 

6572 

6637 

6302 

6917 

7032 

7147 

7262 

7377 

115 

8 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

£410 

8525 

115 

9 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114 

380 

579784 

579398 

580012 

580126 

580241 

580355 

580469 

580583 

580697 

580811 

114 

] 

580925 

581039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

19£0 

114 

2 

206:i 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

3 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

39921  4105 

4218 

113 

4 

4331 

4444 

4557 

4670 

4783 

4396 

5009 

51221  5235 

5348 

113 

5 

5461 

5574 

5636 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

6 

6537 

6700 

6312 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8334 

8496 

8608 

8720 

112 

S 

8332 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

U2 

9 

9950 

590061 

590173 

590284  590396 

590507 

590619 

590730  590342 

590953 

112 

390 

591065 

591176 

5912S7 

591399  591510 

591621 

591732 

591.843  591955 

592066 

111 

1 

2177 

2238 

2399 

2510 

;  2621 

2732 

2343 

2954;  3064 

3175 

111 

2 

3236 

3397 

3503 

3613 

:     3729 

3340 

3950 

4(:6i:  4171 

4232 

111 

3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

51651  6276 

5336 

110 

4 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

5 

6597 

6707 

6317 

6927 

1  7037 

7146 

7256 

7366 

7476 

7586 

i  110 

6 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8631 

1  no  1 

7 

8791 

8900 

9009 

9119   9223 

9337 

9446 

9566 

9665 

9774'  1091 

8 

9333 

9992 

600101 

600210  600319 

,600423  600537 

600646  600755  eO0.-;64  liiy|| 

No 

600972 

601082 

1191 

1299   1403 

1517 
5 

1625 
6 

1734 
7 

1843;  1951 

109 

0 

I   1 

a 

3 

4: 

8   1   9 

Diff 

TABLE    Xll.       LOGAItlTHMS    OF    NUMBEUS. 


161 


I  No.' 
!4U0 

1 

2 

3 

4 

5 

6, 

7i 


0^1  1 

6lv2()6U  602169 
3M4 


4226 

53):') 

6331 
7455 

8526 
9194 


3253 
4334 
54)3 
6439 
7562 
6633 
9701 


S  610660  610767 


a 


9 

410 
I 
2 
3 

4 


1723 

6127S4 
3342 
4897 
5950 
7000 
8043 
9093 

620136 
1176 
2214 


420  623249 


4232 
5312 
6340 
7366 
8339 
6j  9410 

7  63  )ia»5 

8  1444 
2157 


1629 

612390 
3947 
5003 
6055 
7105 
8153 
9193 

620240 
1230 
2313 

623353 
4335 
5115 
6443 
7463 
8491 
9512 

63053:) 
1515 
2559 


602277 
3361 
4442 
5521 
6596 
7669 
8740 
9803 

610373 
1936 


602336 
3469 
4550 
5623 
6704 
7777 
8347 
9914 

610979 
2012 


612996 
4053 
5103 
6160 
7210 
8257 
9302 

620344 
1334 
2421 

623456 

4438 
5518 
6516 
7571 
8593 
9613 


613102 
4159 
5213 
6265 
7315 
8362 
9406 

62()443 
1433 
2525 


602494 
3577 

4658 
5736 
6311 
7834 
8954 
610021 
1036 
2143 

613207 
4264 
5319 
6370 
7420 
8466 
9511 

620552 
1592 
2623 


5 

602603 
3636 
4766 
5344 
6919 
7991 
9061 

610123 
1192 
2254 

613313 

4370 
5424 
6476 
7525 
8571 
9615 
620656 
1695 
2732 


6 


602711 
3794 
4374 
5951 
7026 
8093 
9167 

610234 
1296 

2360 

« 

613419 
4475 


623559  623663 


430  633463 

1   4477 


5434 
6438 
7490 
8439 
9436 
7613131 
8i  1474 
9 I  2455 


440  613453 


4591 
5621 
6643 
7673 
8695 
9715 
630631  630733 
1647   1743 


II 
2i 

4i 

5i 
61 


4439 
5422 
6104 
7353 
8360 
9335 


633569 

4578 
5534 
6533 
7590 
8539 
9536 
640531 
1573 
2563 

643551 
4537 
5521 
6502 
7431 
8453 
9432 


2660 

633670 
4679 


8 

602319  602926 
4010 
50^9 
6166 
7241 
8312 
9331 
610447 
1511 
257-2 


9   Diffi 


7  65030S  650405 


8 
9 

450 
1 
2 
3 
4 
5 
6 
7 


1375 
2343 


1276 
2246 

653213 

4177 

5133 

6093 

7056. 

8011 

8965 

9916  660111 
660365  0960 

1813   1907 


5635 
6633 
7690 
8639 
9636 
64003) 
1672 
2662 

643650 
4636 
5619 
6600 
7579 
8555 
9530 

650502 
1472 
2440 


2761 

633771 
4779 

5735 
6789 
7790 
8739 
978^ 
610779 
1771 
2761 

643749 
4731 
5717 
6693 
767^ 
8653 
9627 

650599 
1569 
2536 


4695 
5724 
675 
7775 
8797 
9317 
630335 
1849 
2362 

633372 
4330 


5529 
6581 
7629 
8670 
9719 
620760 
1799 
2835 


3902 
4982 
6059 
7133 
8205 
9274 
610341! 
1405 
2466 

613525 
4531 
5634 
6636 
7734 
8780' 
9324 

620364 
1903 
2939 


5336 
6339 
7390 
8333 
9335 
640379 
1871 
2360 


623766 
4793 
5 

6853 
7373 
8900 
9919 

630936 
1951 
2963 

633973 
4931 
5936 
6939 
7990 
8933 
9934 

640978 
1970 
2959 


603036 

108 

4116 

103 

5197 

103 

6274 

103 

7313 

107 

8419 

107 

9438 

1(17 

610551 

107 

1617 

106 

2676 

106 

613630 
463G 
5740 
6790 
7839 
8834 
9923 

62096- 
201)7 
3042 


623869 
4901 
5929 
6956 
7930 
9002 

630021 
1033 
2052 
3061 

634074 
5031 
6037 
7089 
8090 
9038 

640034 
1077 
2069 
3053 


623973  621076 
5107 
6135 
7161 
8185 
9206 
630224 
1241 
2255 
3266 


653309 
4273 
5235 
6194 

7152 
8107 
9060 


653405 

4369 
5331 
6290 
7217 
8202 
9155 
660106 
1055 
2002 


613847 
4332 
53 1 5 
6796 
7774 
8750 
9724 

650696 
1666 
2633 


643946 
4931 
5913 
6894 

7672 
8343 
9321 
650793 
1762 


Na  O 


653502 
4465 
5127 
6336 
7313 
8293 
9250 

660201 
1150 
2096 


53593 
4562 
5523 
6482 
7433 
8393 
9346 
66029G 
1245 
2191 


653695 
4653 
5619 
6577 
7534 
8133 
9441 

660391 
1339 
2236 

5 


644044 
5029 
6011 
6992 
7969 
8945 
9919 

650390 
1859 
2326 

653791 
4754 
5715 
6673 
7629 
8534 
9536 

660436 
1434 
2330 


6032 
7053 
8032 
9104 
630123 
1139 
2153 
3165 

63417;:> 
5132 
6137 
7189 
8190 
9183 

640183 
1177 
2163 
3156 

644143 
5127 
6110 
7039 
8067 
9043 

650016 
0937 
1956 
2923 


613736 

4792 
534." 
6395 

794;; 

8989 
620032 
1072 
2110 
3146 

624179 
5210 
6233 
726:j 
82S7 
9306 

630:326 
1342 
2356 
3367 


634276 
5233 
6237 
7290 
8290 
9237 

640233 
1276 
2267 
3255 

644242 


108 

106 

105 

105 

105 

105 

104 

104  I 

1041 

104 

103 

103 
103 
103 
102 
102 
102 
102 
101 
101 


634376 
533: 
6333 
7390 
83^9 
9337 

540332 
1375 
2366 
3354 

644340 


653S83 
4350: 
5310 
6769 
7725 
8679 
9631 

660531 
1529 
2475 


6 


5226 
6208 
7137 
816:)' 
9140 
650113 
1081 
2053 
3019 

653934 
4946 
5906 
6364 
7820 
8774 
9726 

660676 
1623 
2569 

8 


6306. 
7235 
8262 
923 
6.50210 
1161 
2150 
3116 

6540^0 
5042 
6002 
6960 
7916 
8870 
9321 

660771 
1713 
2663 


101 

101 

100 

100 

100 

100 

99 

99 

99 

99 

93 
93 
93 
93 
98 
97 
97 
97 
97 
97 


96 
96 
96 
96 
96 
95 
95 
95 
95 
9^. 

9  iCiff. 


Ib):^ 

TABLE  XII.   LOGARITHMS 

>  OF 

NUMBERS. 

No. 

460 

0  1 

662753 

1  1 
662S52 

3 

3 

■*  1 
663135 

5    6    7     \ 
663230  663324  663418 

8  1 

9 

5636071 

Di£L  ) 

94 

662947 

563041 

663512 

1 

3701 

3795 

3339 

3983 

^078 

4172   42661 

4360 

4454 

4543 

94 

2 

4642 

4736 

4330  4924 

5018 

5112 

5206 

5299 

5393 

6487 

&4i 

3 

5581 

5675 

5769  5862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

4' 

6518 

6612! 

6705   6799 

6392 

6986 

7079 

7173 

7266 

7360 

94 

5 

7453 

7546 

7640 

7733 

7S26 

7920 

8013 

8106 

8199 

8293 

93 

6 

83S6 

8479 

8572 

8665 

S759 

8852 

8945 

9033 

9131 

9224 

93 

7 

9317 

9410 

9:503 

9r:96 

9639 

9782 

9875 

9967; 

670060 

670153 

93 

8 

670246 

670339 

670431 '670524 

670617 

670710.670302  670895 

0938 

1030 

93 

9 

1173 

1265 

1353 

1451 

1543 

1636   1723   1821 

1913 

2005 

93 

470 

672098 

672190 

6722S3 

672375 

672467 

672560  672652  672744 

672836 

672929 

92 

1 

3021 

3113 

3205 

3297 

3390 

3432 

3574 

3666 

3758 

3850 

92 

2 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

45S6 

4677 

4J69 

92 

3 

4361 

4953 

5045 

5137 

5223 

5320 

5412 

6503 

5595 

6637 

92 

p 

4 

577S 

5370 

5962 

6053 

6145 

6236 

6323 

6419 

6511 

6602 

.92^ 

5 

6694 

6785 

6576 

6968 

7059 

7151 

7242 

7333 

7424 

7516 

91 

6 

7607 

7698 

7789 

7381 

7972 

8063 

8154 

8245 

8336 

8427 

91 

7 

8518 

8609 

8700 

8791 

8832 

8973 

9064 

9155 

9246 

9337 

91 

8 

9423 

9519 

9610 

9700 

9791 

9532 

9973 

680063, 

630154 

630245 

91 

9 

630336 

630426 

630517  630607 

680693 

630789 

680879 

0970 

1060 

1151 

91 

'  480 

681241 

631332 

681422 

681513 

681603 

681693 

681784 

681874 

631964 

682055 

90 

!   1 

2145 

22:35 

2326 

2416 

2506 

2596 

2636 

2777 

2567 

2957 

90 

1 

i   2 

3047 

3137 

3227 

3317 

3107 

3197 

3587 

3677 

3767 

3857 

90 

1 

!   3 

3947 

4037 

4127 

4217 

4307 

4396 

44.36 

4576 

4666 

4756 

90 

4 

4345 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

90 

5 

5742 

5331 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

6 

6636 

6726 

6315 

69(M 

69M 

7083 

7172 

7261 

7351 

7440 

89 

7 

7529 

7618 

7707 

7796 

7836 

7975 

8064 

8153 

8242 

8331 

89 

8 

8420 

S509 

8593 

8637 

8776 

8865 

8953 

9042 

9131 

9220 

89 

9 

9309 

9393 

9436 

9575 

9664 

9753 

9841 

9930 

650019 

690107 

89 

490 

690196 

690235 

690373 

690462 

690550 

690639 '690723 

690318 

690905 

690993 

89 

1 

1031 

1170 

12.53 

IM7 

1  1435 

1524   1612 

1700 

1789 

1877 

88 

2 

1965 

2053 

2142 

2230 

2313 

2406  2494 

2533 

2671 

2759 

•68 

3 

2347 

2935 

3023 

3111 

3199 

3237 

a375 

3463 

3551 

3639 

88 

4 

3727 

3315 

3903 

3991 

1  4078 

4166 

42^ 

4342 

4430 

4517 

88 

5 

4605 

4693 

4731 

4363 

4956 

5044 

5131 

5219 

5307 

5394 

88 

6 

5432 

5569 

5657 

5744 

5332 

5919 

6007 

6094 

6162 

6269 

87 

7 

6356 

6444 

6531 

6618 

:     6706 

6793 

63.30 

6963 

7055 

7142 

87 

8 

7229 

7317 

7404 

7491 

^  7578 

7665 

7752 

7839 

7926 

8014 

87 

9 

8101 

8183 

8275 

8362 

8449 

1 

8535  8622 

8709 

8796 

8883 

87 

500 

693970 

699057 

699144 

699231  699317 

699404  699491 

699578 

699664 

699751 

87 

j   ] 

9333 

9924 

700011 

700093  700134 

700271  !700353;70O444 

700531 

700617 

87 

2 

700704 

700790 

0377 

0963   1050 

1136 

1222 

1309 

1395 

1432 

86 

1   3 

1563 

1654 

1741 

1827   1913 

1999 

2036 

2172 

2253 

2ai4 

86 

4 

2431 

2517 

2603 

2639   2775 

2361 

2947 

30a3 

3119 

3205 

86 

5 

3291 

3377 

3463 

3;549   3635 

3721 

3507 

3393 

3979 

4065 

86 

6 

4151 

4236 

4322 

4403   4494 

45791  4665 

4751 

4337 

4922 

86 

7 

5003 

5094 

5179 

5265   5350 

5436 i  5522 

5607 

6693 

1  5778 

86 

8 

5864 

5949 

6035 

6120   6206 

6291 

6376 

6462 

6547 

6632 

85 

9 

6718 

6303 

6338 

6974   7059 

1 

7144 

7229 

7315 

7400 

7485 

85 

510 

707570 

707655 

707740 

707826  707911 

707996 

703031 

703166 

703251 

703336 

85 

1 

8421 

8506 

8591 

S676  8761 

8846!  8931 

9015 

9100 

9185 

85 

2 

9270 

9355 

9440 

9524  9609 

9694!  9779 

9863 

9943 

710033 

85 

3 

710117 

710202 

7102S7 

710371  710456 

710540  710625 

710710  710794 

0379 

85 

4 

0963 

1043 

1132 

1217   1301 

1335;  1470 

1554 

1639 

1723 

84 

5 

1807 

1892 

1976 

2060   2144 

2229;  2313 

2397 

2481 

2566 

84 

6 

2650 

2734 

2313 

2902   2936 

3070   3154 

3238 

3323 

3107 

84 

7 

3491 

3575 

3659 

3742   3326 

39IOI  3994 

4078 

,  4162 

4246 

84 

8 

4330 

4414 

4497 

4531   4665 

4749   4333 

4916 

500C 

5084 

84 

S 
1  So 

5167 

5251 

5335 

5418   5502 

5536  5669 

5753 

5336 

592G 

!  84 

0 

1   1 

3 

3 

4 

5 

i  6 

7 

18   19 

Diff. 

1 

TiiBLE    XII. 


LOGARITHMS    OF    NUMBERS. 


163 


No.| 

0 

i 

3 

3 

4: 

5 
71G121 

» 

7 

8 

9 

716754 

Dili". 

83 

520  716003 

716037 

716170  716254 

716337 

7165041716588  71 6671 

1 

6S3-i 

6921 

7004 

7088 

7171 

7254 

7338 

74211  7504 

7587 

83 1 

2 

7671 

7754 

7837 

7920 

8003 

8036 

8169 

8253   83:56 

8419 

831 

3 

S502 

8585 

8668 

8751 

8834 

8917 

9000 

9083   9165 

9248 

831 

4 

93311  9414 

9497   95S0 

9663 

9745 

9828 

991 1 

9994 

720077 

83 

5 

72!)15y' 720242 

720325  720407 

720490 

720573 

720655 

720733 

720321 

0J03 

83 

6 

09c)ii|  106S 

1151   1233 

1316 

1398 

1481 

1563 

1646 

1728 

82 

7 

1811 

1893 

1975   2058 

2140 

2222 

2305 

2337 

2469 

2152 

82 

8   263t 

2716 

2798   2881 

2963 

3145 

3127 

3209 

3291 

3374 

82 

9   34.-)6 

353- 

362 1   3702 

3784 

3866 

3948 

4030 

4112 

4194 

S2 

530  724276 

724358 

724 MO  724522 

721604 

724685 

724767 

724849 

724931 

725013 

8;g 

1 

5095 

5176 

5258   5310 

5422 

5503 

5585 

5667 

5748 

5830 

82' 

2 

5912 

5993 

6075'  6156 

6233 

6320 

6101 

6483 

6564 

6646 

S'>l 

3 

6727 

6309 

G390:  6972 

7053 

7134 

7216 

7297 

7379 

7460 

811 

4 

7541 

7o23 

77041  7785 

7866 

7948 

6029 

8110 

8191 

8273 

81 ! 

5 

8354 

8135 

8516]  8597 

8678 

8759 

8841 

8922 

9003 

9084 

81! 

G 

916> 

9246 

9327   9403 

9489 

9570 

9651 

9732 

9813 

9893 

81 

7 

9974 

730055 

730136  730217 

73029S 

730378 

730459 

730.540 

730621 

730702 

81 

S 

7307S2 

03G3 

09441  1024 

1105 

1186 

1266 

1347 

1423 

1508 

81 

9 

1589 

1669 

1750   1830 

1911 

1991 

2072 

2152 

2233 

2313 

81 

5-10 

732394 

732-174 

732555  732G35 

732715 

732796 

732376 

732956 

733037 

733117 

80 

1 

3197 

3278 

3353   3138 

3518 

3598 

3679 

3759 

3839 

3919 

80 

2 

3999 

4079 

41 go;  4240 

4320 

4400 

4480 

4560 

4610 

4720 

80 

3 

4S0O 

4S30 

4960:  5010 

5120 

5200 

5-^79 

5359 

5139 

5519 

80 

4 

5599 

5679 

5759;  5833 

5918 

5998 

G078 

6157 

6237 

6317 

80 

5 

6397 

(M76 

6556   6635 

6715 

6795 

6371 

6954 

7034 

7113 

80 

6 

7193 

7272 

7352;  7431 

7511 

7590 

7G70 

7749 

7829 

7908 

79 

7 

79S7 

8067 

8146;  8225 

8305 

8384 

8463 

8543 

8622 

8701 

79 

8 

8781 

8860 

8939   9'I18 

9097 

9177 

9256 

9335 

9414 

949:i 

79 

9 

9572 

9651 

9731   9310 

98S9 

9968 

740047 

740126 

740205 

740284 

79 

550 

7403G3 

740412 

7 10521 1740600 

740678 

740757 

740336 

740915 

740994 

741073 

79 

I 

1152 

1230 

1309!  i:388 

-  1467 

1546 

1624 

1703 

1782 

186'1 

79 

2 

1939 

20!S 

2036 ;  2175 

2254 

2332 

2411 

2489 

2568 

2647 

79 

3 

2725 

2304 

2iSi>     2961 

3039 

3118 

3196 

3275 

3353 

3431 

78 

4 

3510 

35-;: 

3567:  3745 

3323 

3902 

3980 

4058 

413G 

4215 

78 

5 

4293 

4  3/-i 

4 119   4528 

4606 

4684 

476:^ 

4340 

4919 

4997 

78 

G 

5075 

5153 

5231 

5309 

5337 

5465 

5543 

5621 

5699 

5777 

78 

7 

5855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

8 

6634 

6712 

6790 

6363 

6945 

7023 

7101 

7179 

7256 

7334 

78 

9 

7412 

7489 

7567 

7645 

7722 

7800 

7878 

7955 

8033 

8110 

78 

"GO 

748183 

74326G 

743343  748421 

748493 

748576 

748653 

748731 

748808 

74388." 

77 

1 

89G3 

9;>10 

9118 

9195 

9272 

9350 

9427 

9501 

9582 

r;650 

77 

2 

9736 

9811 

9391 

9968 

750045 

750123 

750200 

750277 

750354 

750431 

77 

375050S 

75058G 

750663  750740 

0817 

0894 

0971 

1048 

U25 

12il2 

77 

4 

1279 

1356 

1433   1510 

1537 

1661 

1741 

1818 

1895 

1972 

77 

5 

2018 

2125 

2202   2279 

2356 

2433 

2509 

258G 

2663 

2740 

77 

G 

2S1G 

2^93 

2970   3047 

3123 

3200 

3277 

3353 

343') 

3506 

77 

7 

3583 

3660 

3736  3313 

3339 

3966 

4042 

4119 

4  1 95 

4272 

77 

8 

4313 

4125 

4501 

4578 

4654 

4730 

4807 

4383 

496  I 

5036 

76 

9 

5112 

5189 

5265 

5341 

5417 

5494 

5570 

5646 

5722 

5799 

76 

570 

755375 

755951 

756027 

756103 

756180 

756256 

756332 

756108 

756484 

756560 

76 

I 

G^13G 

6712 

6788 

6361 

6940 

7016 

7092 

7163 

724  1 

7320 

76 

2 

7396 

7472 

7548 

7624 

7700 

7775 

7851 

7927 

80' tt 

8079 

76 

3 

8155 

8230 

8306 

8332 

8458 

8533 

8609 

8685 

8761 

8836 

76 

4 

3912 

8938 

9063;  9139 

9214 

9290 

9366 

9441 

9517 

9592 

76 

5 

9863 

9743 

9819   9391 

9970 

760045 

7G0I2I 

760 1 96 

760272 

760347 

75 

fi 

761122 

760493 

760573  760619 

760724 

0799 

0375 

095!) 

1025 

1101 

75 

7 

1176:  1251 

1326 

1402 

1477 

1552 

1627 

1702 

177^ 

1853 

75 

8 
9 

\:)-i^\     2003 

2078 

2153 

2223 

2303 

2373 

2453 

2529 

2604 

75 

2679 
0 

,  2754 

2329 
3 

29ai 

2978 

3053 
5 

3123 
6 

3203 

7 

3278 
8 

3353 
9 

75 
Dlff. 

Na 

1 

3 

4 

164 


TABLE    Xll.        LOGARITIOIS    OF    NUMBERS. 


No.i 
530; 

0 

76.:«2S 

1  j 
763503 

3 

3  i  *  1 

5 

6 

763877 

7     , 

8 

»  : 

Diff. 

75 

763573  763653  763727 

763302 

763952 

764027:7641011 

1 

4176 

42511 

43261  4400^  4475 

45501  4624 

4699 

4774 

43481 

75 

2 

4923  4993 

5072   5147   5221 

5296:  5370 

5445 

5520 

5594 

75 

3 

5669   5743 

5318   5892   5966 

6041   6115 

6190 

6264 

6333] 

74 

4 

6413   64S7 

6562 

6636   6710 

6785   6359 

6933 

7007 

70821 

74 

5 

7156   7230 

7304 

7379,  7453 

7527   7601 

7675 

7749 

7823 

74 

6 

789S;  7972 

8046   8I20;  8194 

8268 

8342 

8416 

8490 

8564 

74 

7 

S63SI  8712 

8766i  8860   8934 

9008 

9082 

9156i 

9230 

.9303 

74 

8 

9377;  9451! 

9525   9599   9673 

9746 

9820 

98941 

9968 

770fi42 

74 

9 

770115 

770189; 

770263  770336  7704101 

1     1 

770484 

770557 

770631 

770705 

0778 

■  74 

590 

770352 

770926 

770999 

771073  771146 

771220 

771293 

771367 

771440 

771514; 

74 

1 

15S7 

1661 

1734 

1803:  1331 

1955 

202.S 

2102 

2175: 

2243' 

73 

2 

2322 

2395 

2463 

2542   2615 

2688 

2762 

2835 

2908  j 

2981 

73 

3 

3055 

3123 

3201 

3274!  3348 

3421 

3494 

3567 

3640 

3713 

73 

4 

3786 

3360 

3933 

40061  4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736!  4809 

4882 

4955 

5028 

5100 

5173 

73 

6 

5246 

5319 

5392 

5465   5538 

5610 

5683 

5756 

5829 

5902 

73 

7 

5974 

6047 

6120 

61931  6265 

6338 

&41I 

6483 

6556 

6629 

73! 

8 

6701 

6774 

6846 

6919,  6992 

7064 

7137 

7209 

7282 

7354 

1-0 

9 

7427 

7499 

7572 

7644'  7717 

j 

7789 

7862 

7934 

.8006 

8079 

72 

600 

778151 

773224 

778296 

778363  778441 

778513 

778585 

778658 

778730 

778802 

72 

] 

8374 

8947 

9019 

9091  i  9163 

9236 

9303 

9380 

9452 

9524 

72 

2 

9596 

9669 

9741 

9813'  9835 

9957 

730029 

780101 

780173 

780245 

72 

3 

780317  7S0aS9 

730461 

780533  780605 

7S0677 

0749 

0821 

0893 

0965 

72 

4 

1037;  1109.  1181 

1253:  1324 

1396 

1468 

1540 

1612 

1684 

72 

5 

1755 

1827 

1899 

1971   2042 

2114 

2186 

2253 

2329 

2401 

72 

6 

2473 

2544 

2616 

2688,  2759 

2831 

2902 

2974 

3046 

3117 

72 

7 

3139 

3260 

3332 

3403   3175 

3546 

3618 

3689 

3761 

3832 

71 

'   8 

3904 

3975 

4046 

4113   4189 

4261 

4332 

4403 

4475 

4546 

71 

9 

4617 

4639 

4760 

4331   4902 

4974 

5045 

5116 

5187 

5259 

71 

610 

735330 

735401 

785472 

785543  785615 

785686 

785757 

785828 

785899 

785970 

71 

1 

6041 

6112 

6183 

62^54   6325 

6396 

6467 

6533 

6609 

6630 

71 

2 

6751 

6322 

6393 

6964   7035 

7106 

7177 

7248 

7319 

7390 

71 

3 

7460 

7531 

7602 

7673   7744 

7815 

7885 

7956 

8027 

8098 

71 

4 

8168 

8239 

8310 

83S1   8451 

8522 

8593 

8663 

8734 

8804 

71 

5 

SS75 

8946 

9016 

9087   9157 

9228 

9299 

9369 

9440 

9510 

71 

6 

9581 

9651 

9722 

9792  9363 

9933 

790004 

790074 

790144 

790215 

70 

7 

790235 

790356 

790426 

790496  790567 

790637 

0707 

0778 

0848 

0918 

70 

8 

09SS 

1059 

1129 

1199   1269 

laio 

1410 

1480 

1550 

1620 

70 

9 

1691 

1761 

1831 

1901   1971 

2041 

2111 

2181 

2252 

2322 

70 

620 

792392 

792462 

792532 

792602  792672 

792742 

792812 

792^82 

792952 

793022 

70 

1 

3092 

3162 

3231 

330  r  3371 

3441 

3511 

3531 

3651 

3721 

70 

2 

3790 

3860 

39:30 

4000  4070 

4139 

4209 

4279 

4349 

4413 

70 

3 

4483 

455S 

4627 

4697   4767 

4836 

4906 

4976 

5045 

5115 

70 

4 

5185 

5254 

5324 

5393   5463 

5532 

5602 

5672 

5741 

5811 

70 

5 

5830 

5949 

6019 

6038   6158 

6227 

6297 

6366 

6436 

6505 

69 

6 

6574 

6644 

6713 

6732   6352 

6921 

6990 

7060 

7129 

7198 

69 

7 

7268 

7337 

7406 

7475   7545 

7614 

7683 

7752 

7821 

73S0 

69 

8 

7960 

8029 

8093 

8167   8236 

8305 

8374 

8443 

8513 

8582 

69 

9 

8651 

8720 

8789 

8858   8927 

8996 

9065 

9134 

9203 

9272 

69 

630 

799341 

799409 

799478 

799547  799616 

799685 

799754 

799823 

799892 

799661 

69 

1 

3U0029 

30009- 

3001671300236  800305 

800373 1800442 

800511 

800530 

800643 

69 

2 

0717 

0786 

0354 

0923   0992 

1061 

1129 

1198 

1266 

1335'  69 

3 

1404 

1472 

1541 

1609   1673 

1747 

1815 

1834 

1952 

2021 

69 

4 

2aS9 

2153 

2226 

2295  2363 

2432 

2500 

2563 

2637 

2705 

03 

5 

2774 

2842 

2910 

2979   3047 

3116 

3184 

3252 

3321 

3389 

CS 

6 

34571  3525 

3594 

3662   3730 

3793 

3S67 

3935 

:  4003 

4071 

68 

7 

4139 

4203 

4276 

4344  4412 

4430 

4548 

4616 

4635 

4753 

68 

c 

432 1 

4339 

4957 

5025   5093 

5161 

5229 

5297 

!  5365 

5433 

63 

ft 

5501 

5569 

1 

5637 

5705   5773 

5841 

5908 

5976 

6044 

j 

6112 

1  68 

Na 

0 

2 

3    4 

5 

6 

7 

1  8 

9 

Diff. 

TABLE    Xll.       LOGARITHMS    OF    NUMBERS. 


165 


No. 


640 
1 

2 
3 

4 
5 
6 
7 
8 
9 


8061801806218 


6S58 
7535 
8211 
8SS6 
9560 
810233 
0904 
1575 
2245 


69  6 
7603 
8279 
8953 
9327 
810300 
0971 
1612 
2312 


650312913 


8 
9 

660 
1 
2 
3 
4 
5 
6 
7 


3581 

424S 

4913 

5578 

6241 

6904 

7565 

8226 

8885 


800:316 
6994' 
7670 
8346 
9021 
9694 

810367 
1039 
1709 
2379 


812980 
3648 
4314 
49S0 
5644 
6303 
6970 
7631 
82:)2 
8951 


819544 
820201 
0358 
1514 
2163 
2322 
3474 
4126 
4776 
5426 


819610 
320267 
0924 
1579 
2233 
23S7 
3539 
419! 
4841 
5491 


806451 
7129 
7806 
84SI 
9156 
9829 

810501 
1173 
1843 
2512 


806519 
7197 
7873 
8549 
9223 
9896 

810569 
1240 
1910 
2579 


813114 
3781 
4447 
5113 

5777 
6440 
7102 
7764 

8424 
9083 


IG70 
1 
2 
3 
4 
5 
6 


81 9676 
320333 
0989 
1645 
2299 
2952 
3605 
4256 
4906 
5556 


813181 
3348 
4514 
5179 
5843 
6506 
7169 
7830 
8490 
9149 


826075 
6723 
7369 
8015 
8660 
9304 
9947 

830539 
1230 
1870 


819741 
829399 
1055 
1710 
2364 
3013 
3670 
4321 
4971 
5621 


806655 
7332 
8008 
8634 
9358 

810031 
0703 
1374 
2044 
2713 


81324 
3914 
4531 
5246 
5910 
6573 
7235 
7896 
8556 
9215 


_^1 
806723 
7400 
8076 
8751 
9425 
810093 
0770 
144 
211 
2780 


9   Diff 


813314 
3981 
4647 
5312 
5976 
6639 
7301 
7962 
8622 
9281 


806790 
7467 
8143 

8818 
9492 
810165 
0837 
1508 
2178 
2847 


819807 
820464 
1120 
1775 
2430 
3083 
3735 
43S6 
5036 
5636 


813381 

4048 
4714 
5373 
6042 
6705 
7367 
8028 
8638 
9346 


819873 
820530 
1186 
1841 
2495 
3148 
3300 
4451 
5101 
5751 


813448 
4114 
4780 
5445 
6109 
6771 
7433 
8094 
8754 
9412 


826140 
6787 
7434 
8030 
8724 
93681 

330014 
0653 
1294 
1934 


826204 
6352 
7499 
8144 
8789 
1  9132 
330075 
0717 
1358 
1998 


6SC 
I 

2 
3 
4 
5 
6 
7 
8 
9 


832509 
3147 
3734 
4421 
5056 
5691 
6324 
6957 
7533 
8219 


826269 
6917 
7563 

8209 
8853 
9497 
830139 
0781 
1422 
2062 


832573 
3211 
3343 

4434 
5120 
5754 
6337 

7020 
7652 

8232 


82G334 
6931 
7628 
8273 
8918 
9561 

830204 
0845 
1486 
2126 


832637 
327." 
3912 
4548 
5183 
5317 
6451 
7033 
77  Ir 
8345 


819939 

820595 
1251 
1906 
2560 
3213 
3865 
4516 
5166 
5815 


813514 
4181 
4347 
5511 
6175 
6833 
7499 
8160 
8320 
9478 


820004 
0661 
1317 
1972 
2626 
3279 
3930 
4581 
5231 
5880 


820070 
0727 
1332 
2037 
2691 
3344' 
3996 
4646 
5296 
5945 


826399 
7046 
7692 
8333 
8982 
962 

830263 
0909 
1550 
2189 


832700 
3333 
3975 
4611 
5247 
5831 
6514 
7146 
7773 
8403 


326464 
7111 
7757 
8402 
9046 
9690 

830332 
0973 
1614 
2253 


832764 
3402 
4039 
4675 
5310 
5944 
6577 
7210 
7841 
8471 


326528 
7175 
7821 
8467 
9111 
9754 

830396 
1037 
1678 
2317 


832323 
3466 
4103 
4739 
5373 
6007 
6641 
7273 
7904 
8534 


b90;  833849 
1 1  9178 
21840106 


0733 
1359 
19S5 
2609 
3233 
3355 
4477 


833912 
9.;  41 

840169 
0796 
1422 
2047 
2672 
3295 
3918 
4539 


ilwo.    O 


833975: 
96041 

840232 
0359 
1435 
2110 
2734 
3357 
3980 
4691 

3 


83903S 
9667 

840294 
0921 
1547 
2172 
2796 
3420 
4042 
4664 


839101 
9729 

840357 
0934 
1610 
2235 
2859 
3482 
4104 
4726 


820136 
0792 
1448 
2103 
2756 
3409 
4061 
4711 
5361 
6010 


826593 
7240 
7886 
8531 
9175 
9818 

830460 
1102 
1742 
2381 


832892 
3530 
4166 
4302 
5437 
6071 
6704 
7336 
7967 
8597 


39164 
9792 
840420 
1046 
1672 
2297 
2921 
3544 
4166 
4788 


826658 
7305 
7951 
8595 
9239 
9882 

8305 
1166 
1806 
2445 


63 

63 
63 
67 
67 
67 
67 
67 
67 
67 

67 
67 
67 
66 
06 
66 
66 
6G 
66 
66 

66 
€6 
66 
65 
65 
05 
65 
65 
65 
65 

65 
65 
65 
64 
64 
64 
64 
64 
61 
64 


832956 
3593 
4230 
4866 
5500 
6134 
6767 
7399 
8030 
8660 


833020 
3657 
4294 
4929 
5564 
619 
6830 
7462 
8093 
8723 


839227 
^55 

840482 
1109 
1735 
2360 
2983 
3606 
4229 
4850 


839239 
9918 

840545 
1172 
1797 
2422 
3046 
3669 
4291 
4912 


833083 
3721 
4357 
4993 
5627 
6261 
6394 
7525 
8156 
8786 


8 


839352  839415 

9931 
840608 

1234 

1860 

2484 

3108 

3731 

4353 

4974 


166 


TABLE    XII.       LOGARITHMS    OF    KX'3IBEBS. 


Sa      O 


6 


8 


700  S450&S  Sioien  S45222  S452S4  S45:i46 
ll     571 S      5rS<J      5>i2      5&W      59G6 
63&?      &i6l      6523     6565' 


S454C«S  54547'   ^-"'^2  S45594  S 


6337 
6955 
7573| 
SI  59 


7017, 

7684 

S251 


S505;     SS66 


7tJ79  7141  7202 

7696  775S  7519 

8312*  S374  S435 

S92S  S^9  9051 

o-jo  ci«wi  fy;* 


710^ri5S  351320  5513=1  = 


1S70      1931 
■24S»     ^5*1 


3695 

4306 
4913! 
55191 


3l5Ci 

3759 
4367 
4974 
555»:» 
6124'  61S5 
6729      67S9 


If-  - 

2&    4 

3211 
3S2.J 
4425 

5*  - 
56-. 
6245 
6S5'J 


....": 


3272 
35-51 


6910 


6025 
6646 
7264 
75i51 
84£'7 
9112 


C-l-AJ 


6fr 

67l- 
7326 
7943 

S559 
9174 


:  6213 

^77u  6532 

73SS  7449 

8004  S066 

8620  8682 

9235  9297 

F^9  9911 

"   -  55C«524 

- -c  1136 


S4?656 
6275 
6594 
7511 
81 2S 
8743 
9355 
9972 

55C555 
1!&7, 


DiftM 

i 


3333     3394 
39411 


6970 


2236 
2546 
3455 


2297 


23r5 

2965 
3577 


2419 
3029 
£6?7 


;;-—:» 


-o  o  =.^74.53!s57513'S57574 
5»:66   -5116.  5176 


71  15341  1594  1654 
81  2131!  2191  5251 
9      2725      2757.     2547  j 


Ai'.  : 

-   ■•--> 

4124 

4!n5 

424 

5 

4- 

'.1 

473! 

47i-2 

jc; 

0 

.' ,  - 

64^7 
7t>3l 

64  ■r7 
7091 

OO-IS 

7152 

66i«5 
7212 

C' 

<^- 

- 

S57fi"-' 

i-.-c:i.< 

^r"*  .   .  r>-* 

..-.,. 

c -",*;" 

5.. 

- 

^ 

I 

iiiO  1176.  r^j<j  li   '  .      " 

1714,  1773!  153:3  1^.  -     -  , 

2310,  2370  2430  245v  254yi  26tK? 

2906,  29661  3025;  3055;  3144,  3204 


2665 
3263 


llr. 


3 

-:0  1  1 

5104 

516:3 

5-^ 

52-2 

4, 

5755 

o^   - 

"■"*■' 

D 

6257 

6346 

64 

6 

6575 

e9:r 

&- 

.    .- 

^ 

7i«~ 

"?^*>^ 

75  r 

~ 

5174 

5i3o 

5762 

S521 

_  _: 

,1  ;/^->:;p,  Qfi'w^'  =636=0  863739  563799' S63555 
4274   4333   4392:  4452 

-.:-.-   ^-r   4567   4926   49S5J  f'"-" 
5-541   5400   5459   5519   5575'  c- 


7521   75''0   79?? 

54    -  -     ■ " - " 

S521   5579,  S&S5,  a^. 


74/,  i^ 


-r>"-*-y  «..~  J-. 


a-'i^  ^;?-J 


!        I       "i        I    '"'1     "i 

fe  ^Q-'.-  ico^^Li  vcpjv  c>'0'r.r;pQ-rj-i 


-X,   Ig^o,   lt>o^ 
Si  2?56'  2215 


I 


25->5 
Q10- 


iii*,ii, 
^^"'j' 


I, 


5«i  j 


25(^6 
3055; 


2564   26221 

3146   3204 

^07   crx:  = 


730  875061  875119  875177  875235  575293  875351  S75409 

]i  5640  5695   5756 


5640 
^15 
6795 


2 

51 
6 
7 
8 
9 

Ko       O 


5695 
G276 
6553 


69101 


552^ 

■557^ 

-6:37 

9096 

915:3 

921' 

9669 

9726 

9: 

550242 

551299  350::" 

5513 
^91 
6^5 
7544 
5119 
5694 


5571 
6449 
7026 
76f^ 
5177 
5752 


5929 
6507  i 
7053* 

7659 
8234 

55'!'9 


62 

62! 

62 

62 

62 

62 

61 

61 

61 

61 


=51625551656  S51747'S51809!    61 


61 

611 

61! 

6II 
61" 


GO 
60 
60 
60 

59 

59 


59 
59 


-  :  _  . 

^~  "^ - 

-.  -- 

■-  -    - 

5409 

87.=466 

S75524 

875.=, 

..'! 

5957 

6045 

6102 

6 

6564 

6622 

6650 

6:.: 

.'.- 

7141 

7199 

7256 

7314 

58; 

7717 

7774 

7532 

7559 

58i 

^292 

049 

6407 

8464 

57  j 

8566 

=924 

895! 

9CS9 

01  ■■ 

944*^' 

9497 

95  ~." 

f^?!2 

571 

6 


Diff. 


TABLE    XII.       LOGABlTHi: 


NUMBERS. 


161 


2! 

4; 

6; 

S; 
9 


0_ 

ida5\ 

l&55i 

3*51! 

i7ii5 
5361 


I/ii 


144-2 
2)JI2 
25^1 
3i->J 
371S 
42>5 
4>52 
541S 
5953 


14: 


at 

3-. 

37: 

43-^ 

4^.>9 

5474 

603d 


~ll-56^r.- 
1727       ." 
2^7      - 
•25j->*      -2:- 


4i(e"i; 

5531 


50-^1 
61-52' 


3'-i7- 


4->.>' 
5135 
57aj 
G265 


5?1:<2 

57-57 
6321 


KS7; 


^34     56 1 


-47  SS66''4  ssScfifX  55671^: 


5  S»2i     9355      9- 

6  ;  ~      - 

7  H  -    -  _ 


7955      9311'     =*:«7 
S516      -""       --^ 

?:'77 


1705^     I7e&i    ISlSi     lS7:i     1225:     laii:? 


.£>:*      s^ 


175?  5 


3 
4 
5 
6 
7 

9 


- 

-   - 

37621 

S517 

431^' 

».-^—  - 

45: 

-    - 

.54i- 

;--.    - 

-5375 

&iX> 

65-26 

65?  i 

7077 

7132 

^w-%— r»  ~-~ 

1  ^  59-^^  59^2 


-^:^-  j 


6^36 


79f";  597^7 '^765-2 
1      Si 

3    S7: 


r297i     7352:     7iJ7      7-iS^     ■> 


tax 


•  r- 

14 
2- 

--    - 

27641     *5iS'     2S73     29271     :^?. 


3     471t 


3i     a 


9^     7949i     5j»2 


1      ' 
■2      f 


Sf"fc%5      -ii-      5163;     3217;     8270     53^^ 

1 
^j5699. 


o 
6 


9 


11-5? 

16^1' 

2753 
3254 

O 


1-. 

17-. 

2*75 

2506 

.3:»7 


i4-2j 

.:.;  .r:  ..--.  19^ 

23a^  2351,  ^33  24S5 

2559  291^:  2966  ^319 

3391.V  3143  3496  3549 

3 


36/2      :je'-5 
6  7 


3.<.«? 


-i.Ci; 


Mt' 


168 


TABLE    Xll.       LOGARITHMS    OF    NUMBERS. 


No. 

0 

1 
913567 

3 

3 

913973 

4t 

5 

6 

914132 

t 

8 
9142-37 

9 

DiB.] 

820:913314 

913920 

914026 

914079 

914134 

914290 

53  1 

1 

4ai3 

4396   4449 

4502 

4555 

4603 

4660 

4713 

4766 

4819 

53' 

2 

4S72 

492r   4977 

5030 

5033 

5136 

5139 

5241 

5294 

5347 

53  i 

3 

5400 

545b   5505 

55-53 

5611 

5664   5716 

5769 

5822 

5375 

53  1 

4 

5927 

5930   60a3 

6035 

6138 

6191   6243 

6296 

6349 

6101 

53  : 

5 

fr454 

6-507 

6559 

6612 

6G64 

6717   6770 

6322 

6875 

6927 

£3' 

6 

6930 

7033 

7035 

7135 

7190 

7243 

7295 

7343 

7400 

7453 

53;  i 

7 

7506 

7553 

7611 

7663 

7716 

7763 

7320 

7373 

7925 

7973 

52: 

8 

8030 

8083 

8135 

8163 

8240 

3293   8345 

8397 

8450 

~  8502 

52 

9 

8555 

8607 

8&59 

8712 

8764 

8316   8369 

8921 

8973 

9026 

52 

830 

919073 

919130 

919183 

9192-35 

919237 

919310  919392 

919444 

919496 

919549 

62 

1 

9601 

9653 

9706 

975-3 

9310 

9562   9914 

9967 

920019 

920071 

52 

2 

920123 

920176  92022.S 

920230 

920-332 

920334  920136 

920439 

0-541 

0593 

52 

3 

0645 

0697 

0749 

0301 

0353 

09061  09-53 

1010 

1062 

1114 

52 

4 

1166 

1213 

1270 

1322 

1374 

1426 

1473 

1530 

1532 

1634 

52 

5 

1656 

1735 

1790 

1342 

1S94 

1946 

1993 

2050 

2102 

21.54 

52 

6 

2206 

22-53 

2310 

2362 

2414 

2466 

2513 

2570 

2622 

2674 

52 

7 

2725 

2777 

2329 

2331 

2933 

2935 

3037 

3039 

3140 

3192 

52 

8 

3244 

3-296 

3345 

3399 

3451 

3-j03 

355-5 

3607 

3658 

3710 

52 

9 

3762 

3314 

3365 

3917 

3969 

4021 

4072 

4124 

4176 

4223 

52 

840 

924279 

924-331 

924383 

9^44-31 

924486 

924533 

924539 

924641 

924693 

924744 

52 

1 

4796 

4343 

4399 

4951 

5003 

5a54 

5R6 

5157 

5209 

5261 

52 

2 

5312 

5364 

5415 

5-167 

5513 

5570 

5821 

5673 

5725 

5776 

52 

3 

5828 

5379 

5931 

5932 

6034 

6035 

6137 

6188 

6240 

6291 

51 

4 

6342 

6394 

6445 

6497 

6543 

6600 

6651 

6702 

6754 

6305 

51 

5 

6857 

6903 

6959 

7011 

7062 

7114 

7165 

7216 

7263 

7319 

51 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7673 

7730 

7781 

7332 

51 

7 

7833 

7935 

7936 

8037 

8033 

8140 

8191 

8242 

8293 

8345 

51 

8 

8396 

8447 

8493 

S549 

S601 

36-52 

8703 

8754 

aso5 

8357 

51 

9 

8903 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9363 

51 

850 

929419 

92^70 

929521 

929572 

929623 

&29674 

929725 

929776 

929827 

929379 

51 

i 

9930 

9931 

930032 

930033 

930134 

930135 

930236 

930287 

930333 

930339 

51 

2 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0347 

0393 

51 

3 

0949 

lOOO 

1051 

1102 

1153 

1201 

1254 

1305 

1356 

1407 

51 

4 

14-53 

1509 

1560 

1610 

1661 

1712 

1763 

1314 

1365 

1915 

51 

5 

1966 

2017 

2065 

2113 

2169 

2220|  2271 

2322 

2372 

2423 

51 

6 

2474 

2324 

2575 

2626 

2677 

2727   2773 

2329 

2S79 

29-30 

51 

7 

2931 

3031 

3032 

31-33 

3133 

32-31   3235 

3335 

3336 

3137 

51 

8 

»137 

a533 

3539 

3639 

3690 

37401  3791 

3341 

3392 

3943 

51 

9 

3993 

4014 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4443 

51 

860 

934498 

93i549  931599 

931650 

931700 

934751 

931801 

934352  934902 

931953 

50 

J 

5003 

5054 1  5104 

5154 

5205 

5255 

5306 

5356   54C6 

5457 

50 

•2 

5507 

55.53   5603 

56-53 

5709 

5759 

5309 

5360  5910 

5960 

50 

3 

6011 

60611  6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

50 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6315 

6365 

6916 

6S66 

50 

5 

7016 

7066 

•7117 

7167 

7217 

7267 

7317 

7367 

7413 

7463 

50 

6 

7518 

7568 

7613 

7663 

7718 

7769 

7819 

7369 

7919 

7S69 

50 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

3470 

50 

S'  8520 

8570'  8620 

3670 

3720 

8770 

8320 

8870 

8920 

6970 

50 

9|  9020 

9070   9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

50 

870 

939519 

9.39-569  939619 

939669 

939719 

939769 

939319 

939869  939918 

939963 

50 

1 

i«0018 

W0063  9401 13 

940165 

940213 

940267  940317 

940367  940417 

940467 

50 

2 

0516 

0566  0616 

0666 

0716 

0765 

0315 

0565  0915 

0964 

50 

3 

1014 

1061 

1114 

1163 

1213 

1263 

1313 

1362   1412 

1462 

50 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

13-59   1909 

19.58 

50 

5 

2003 

2058 

2107 

21-57 

2207 

2256 

2306 

2355   2405 

2455 

50 

6 

2504 

2-554 

2603 

2653 

2702 

2752 

2301 

■  2S51   2901 

29-50 

50 

7 

3000 

3049   3099 

3143 

3193 

3247 

3297 

3-316  3396 

3145 

49 

8 

3495 

3-544  3-593 

3643 

3692 

3742   3791 

3841   3390 

3939 

49 

9 

3939 

40-3.3  4033 

1  4137 

1  3 

4136 

4236  4235 

4335  4.384 

44-33 

49 
Diff. 

No. 

0 

1   1 

i  3 

4: 

5 

6 

7    8 

9 

TABLE    XII.       LOGARITHMS    OF    x\U3IBtKS. 


169 


No, 

550 
1 
2 
3 
4 
5 
6 
7 
8 
9 


O 

9444S3 
4976 

."-169 
5961 

mo2 

6'J43 
7434 
7924 
>413 
3902 


S90  949390 

H  9S7 


502.3 
551- 
6'Jic 

e^n 

6992 
7433 
7973 
-4^2 
8951 

t4r43- 
9926 


a 


8 


Di£f. 


9443-1 
5074 
5567 
6059 
6551 
7041 
7532 
8022 
8511 
8999 

949485 
9975 


2  950363  950414  950462 


0S51 
13:?S 
1S23 
230S 
2792 
3276 
3760 


9ao'954243 


6' 


0900 
13-6 
1372 
2356 
2341 
3325 
3303 


0949 
143: 

1920 
24  13 1 
2339 
3373 
3356 


944631 
5124 
5616 
(•103 
6600 
7090 
7531 
8070 
8560 
9048 

949536 
950024 
051 1 
0997 
14S3 
1969 
2453 
293S 
3421 
3905 


9446-^! 
5173 
5665 
6157 
C649 
7140 
7630 
8119 
8609 
9097 


944729  944779  94432 
5222] 
5715| 
6207 
6693 
7189 
7679 
8163 
8657 
9146 


5272 
5764 
6256 
6747 
7233 
7728 
8217 
87(16 
9195 


5321 
5313 
6305 
6796 

7287 
7777 
8266 
8755 
9244 


954291  954339 


4725 
5207 
5633 
6163 
6649 
7125 
7607 
8036 
8564 


4773 
5255 
5736 
6216 
6697 
7176 
7655 
8134 
8612 


4321 
5303 
5784 
6265 
6745 
7224 
7703 
8131 
8659 


949535 
950073 
0560| 
1046! 
1532 
2017 
2502 
2936 
3470 
3953 


949634  919683 
950121  950170 

0603!  0657 

1095' 

1530, 

2066 


114 

1629 

•.ell! 


2550 !  2599 


954337  954435 


910  959011  959039  959137 
];  951^1  9566!  96!4 
2   9995  960042  960090 


3  960471 


4 

i 

7 
8' 
9j 

920 

ll 
2' 
3 
4 
5 
6 
7 
8 
9 


0946 
1421 
1395 
2:369 
2343 
3316 

963733 
425  ) 
4731 
5202 
5672 
6142 
6S11 
7<i30 
7543 
8016 


0513 

0994 

1469 

1943' 

2417 

2890 

3363 

963335 

4307 

477S 

52491 

5719! 

6139j 

6653 1 

71-27 

7595 

8062 


0566 
1011 
1516 
199D 
2464 
2937 
3410 


4369 

5a5i 

5832 
6313 
6793 
7272 
7751 
8-229 
8707 

959185 
9661 

960133 
0613! 
1039 
1563 
2033 
2511 
2935 
3457 


4918 
5399 
5380 
6361 
6310 
7320 
7799 
8277 
8755 


3034 
a513 
4001 

95^1434 
4966 
5447 

5923 
6409 
6333 
7363 
7347 
8325 
8303 


3083 
3566 
4049 

954532 
5014 
5495 
5976 
6457 
6936 
7416 
7894 
&373 
8850 


949731 
950219 
0706 
1192 
1677 
2163 
2647 
3131 
3615 
4093 


944377 
5370 
5362 
6354 
6345 
7335 
7326 
8315 
8304 
9-292 

9497S0! 

953267 
0754 
1240 
1726 
2211 
2696 
3180 
3663 
4146 


939-232 

9709 

960133 

06GI 

li:i6 

1  1611 

'  20-5 

2559 

21)32 

3504 


9493-29 
950316 
0303 
1-289 
1775 
2-260 
2744 
3223 
3711 
41^ 


930!  963433 
8950 
9416 
93-2 
970317 
0812 
1-276 
1740 
2203 
2666 


96333'^ 
4354 
43-25 
5296 
5766 

i  6236 
6705 
7173 
7642 
8109 


959230 
9757', 

960233 
0709! 
11341 
1653 
2132 
2606 
3079 
3552 


9593-28 
9304 

960230 
0756 
1231 
1706 
2180 
2G53 
3126 
3599 


963929 
4401' 
4372 
5343 
5313 
62^3 
6752 
7-220 
7633 
8156 


963977 
4443 
4919 
5393 
5-60 


9S3530 
89951 
9463; 
9923' 

970393 
0353; 
1322 
1736' 
-2249, 
2712 


954530 
5062 
5.543 
6024 
6505 
6934 
7464 
7942 
8421 
8393 

959375 
9S52 

9603-28 
0304 
1279 
1753 
2227 
2701 
3174 
3646 


95462 
5110 
5592 
6072 
6553 
7032 
7512 
7990 
8463 
8946 


959423  959471 

9900  9947 

960376  960423 


N<».!     0 


963576 
90431 
9509 
9S75 

970440 
0901 
i:369 
1332 
-2295 
2753 

3 


9636-23 
9090 
9556 

970021 
04-6 
0951 
1415 
1879 
2:342 
a304 


6329 
6799 
7-267 
7735 
8203 

963670 
9136 
9602 

970063 


964024 
4495 
4966 
5437 
5907 
6376 
6345 
7314 
7782 
8249 

963716 
9133 
9649 


9&4071 
4542 
5013 
5434 
5954 
6423 
6392 
7361 
7S29 
8296 


964113 

4590 
5061 
5531 
6001 
6470 
6939 
740S 
7875 


0351 
1326 
ISOl 
2275 
2743 
3221 
3693 


0599 
1374 
1343 
2322 
2793 
3-263 
3741 


964165  964212 


8343  8390  8436  47 


963763 
9-2-291 
96951 
970114;970I61 
05791  0626: 


0997 
1461 
19-25 
2-333 
2351 


1  (44! 
1.50S 
1971 
2434 
2397 


1090, 
1554 
20131 
2431 
2943 


963310 
9276 
9742 

970207 
0672 
1137 
1601 
2064 

1  252 
2939 


4637 
5103 
557S 
6043 
6517 
6936 
74^54 
7922 


4634 
5155 
56-23 
6095 
6364 
7033 
7501 
7969 


963355 
9323 
9739 

970254 
0719 
1183 
1647 
2110 
2573 
3035 

8 


963903 
9:369 
9335 

,970:300 
0763 
1-229 
1693 
2157 
2619 
3032 


43 
43 
43 
43 
43 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 


47 
47 
47 
47 
46 
46 
46 
46 
46 
46 


Diff. 


170 


TABLE    XII.       LOGARITHMS    OF    NUMBERS. 


977724 
8181 
8637 
9093 
9548 

980003 
0458 
0912 
1366 
1S19 


5426 
5875 
6324 


936772 
7219 
7666 
8113 
8559 
9005 
9450 
9895 

990339 
0783 

991226 
1669 
2111 
2554 
299.5 
3436 
3377 
4317 
4757 
5196 


K3. 


0 


1 

1   ^ 
973220 

3  !  4 

973174 

9/3266 

973313 

3636 

36S2 

372>S 

3774 

4097 

4143 

4189 

4235 

4553 

4604 

4650 

4696 

5018 

5064 

5110 

5156 

5478 

5524 

5570 

5616 

5937 

6983 

6029 

6075 

6396 

6142 

6488 

6533 

6354 

6900 

6946 

6992 

7312 

7358 

7403 

7449 

977769 

977815 

977361 

977906 

8226 

8272 

8317 

8363 

8683 

8728 

8774 

8819 

9133 

9184 

9230 

9275 

9594 

9639 

9635 

9730 

930049 

9S0094 

930140 

930185 

0503 

0549 

0594 

0640 

0957 

1003 

I04S 

1093 

1411 

1456 

1501 

1547 

1864 

1909 

1954 

2000 

982316 

932362 

982407 

982452 

2769 

2814 

2.359 

2904 

3220 

3265 

3310 

33L6 

3671 

3716 

3762 

3807 

4122 

4167 

4212 

4257 

4572 

4617 

4662 

4707 

5022 

5067 

5112 

5157 

5471 

5516 

5561 

5606 

5920 

5965 

6010 

6055 

6369 

6413 

6458 

6503 

936817 

986361 

986906 

9S6951 

7264 

7309 

7353 

7393 

7711 

7756 

7800 

7845 

8157 

8202 

8247 

8291 

S604 

8643 

8693 

8737 

9049 

9094 

9138 

9183 

9494 

9539 

9583 

9623 

9939 

9983 

990023 

990072 

9903S3  99042S 

0472 

0516 

0827 

0871 

0916 

0960 

991270 

991315 

991359 

991403 

1713 

1753 

1802 

1846 

2156 

2200 

2244 

2288 

2593 

2642 

26S6 

2730 

3039 

3083 

3127 

3172 

34S0 

3524 

3563 

3613 

3921 

3965 

4009 

4053 

4361 

4405 

4449 

4493 

4S01 

4845 

4889 

4933 

5240 

5284 

5328 

5372 

995679  995723 

995767 

995311 

6117  6161 

6205 

6249 

6555   6599 

6643 

6637 

6993   7037 

7030 

7124 

7430   7474 

7517 

7561 

7867   7910 

7954 

7998 

8303  8347 

8390 

8434 

8739  8732 

8826 

8869 

9174   9218 

9261 

9305 

9609 

9652 
3 

9696 
3 

9739 

1 

4 

5 

6 

973359  973405 

3-20 

3t66 

4281 

4327 

4742 

4788 

5202 

5248 

5662 

5707 

6121 

6167 

6579 

6625 

7037 

7083 

7495 

7541 

977952 

977993 

8409 

8454 

8865 

8911 

9321 

9366 

9776 

9321 

980231 

980276 

0635 

0730 

1139 

1134 

1592 

1637 

2045 

2090 

932497 

982543 

2S49 

2994 

3401 

3446 

3852 

3897 

4302 

4347 

4752 

4797 

5202 

5247 

5651 

5696 

6100 

6144 

6548 

6593 

986996 

987010 

7443 

7488 

7890 

7934 

8336 

8381 

8732 

8826 

9227 

9272 

9672 

9717 

990117 

990161 

0561 

0605 

1004 

1049 

991448 

991492 

1890 

1935 

2333 

2377 

2774 

2819 

3216 

3260 

3657 

3701 

4097 

4141 

4537 

4581 

4977 

5021 

5416 

5460 

995354 

995398 

6293 

6337 

6731 

6774 

7163 

7212 

7605 

7648 

8041 

8035 

8477 

8521 

8913 

8956 

9348 

9392 

9783 

9826 
6 

5 

7           8   1   9 

DifF. 

973451  973497  973543 

46 

3913   3959 

4^05 

46 

4374 

4420 

4406 

46 

4834 

4880 

4926 

46 

5294 

53-10 

53;>o 

16 

6753 

5799 

5346 

46 

6212 

6258 

6304 

46 

6671 

6717 

67(j3 

ir, 

7129 

•  7175 

7^21) 

46 

7686 

7632 

7678 

46 

978043 

978089 

978.3;, 

40 

8500 

8546 

8591 

46 

8956 

9002 

9047 

IG 

9412 

9457 

9503 

46 

9367 

9912 

9958 

46 

950322 

980367 

930412 

15 

0776 

0821 

ose: 

45 

1229 

1275 

1320 

45 

1633 

172,3 

1773 

15 

2135 

2181 

22"G 

45 

982588 

982633 

9S2678 

10 

3040 

3085 

3i:-!P 

45 

3491 

3536 

3531 

45 

3942 

3987 

4032 

45 

4392 

4437 

44,-^ 

45 

4842 

4887 

4932 

45 

5292 

5337 

63S2 

45 

5741 

5786 

6«;^" 

45 

6189 

6234 

6279 

45 

6637 

6682 

6727 

45 

987035 

987130 

987175 

45 

7532 

7577 

7622 

45 

7979 

8024 

800r! 

■15 

8425 

8470 

8514 

45 

8371 

8916 

8960 

45 

S316 

9361 

9405 

45 

9761 

9306 

98oO 

44 

9S0206 

990250 

990294 

44 

0650 

0694 

073S 

41 

1093 

1137 

Ii5^ 

44 

991536 

991580 

991625 

44 

1979 

2023 

2067 

44 

2421 

2465 

2509 

44 

2863 

2907 

29^1 

14 

3304 

3348 

?o':Z 

±4 

3745 

3789 

3833 

44 

4185 

4229 

42"3 

14 

4625 

4609 

4-'-: 

■A 

5065 

5108 

5152 

44 

5504 

5547 

5591 

44 

995942 

S959S6 

996030 

44 

6330 

6424 

6468 

44 

6818 

6862 

69.. i^ 

•  4 

7255 

7299 

7343 

44 

7692 

7736 

7779 

44 

8129 

8172 

82:  ;i 

»1 

8564 

8608 

fc6u2 

44 

9000 

9043 

9087 

44 

9435 

9479 

95'32 

.4: 

9870 

9913 

995, 

DiCfj 

7 

8 

9 

TABLE    X 1 1 1 . 


LOGARITHMIC   SINES,   COSINES,   TANGENTS. 


AND 


0 


OTANGENTS. 


172  TABLE    XIII.       LOGARITHIVIIC    SINES, 


NOTE. 

The  table  here  given  extends  to  minutes  only.  The  usual  methcd 
of  extending  such  a  table  to  seconds,  by  proportional  parts  of  the 
difference  between  two  consecutive  logarithms,  is  accurate  enough 
for  most  purposes,  especially  if  the  angle  is  not  very  small.  When 
the  angle  is  very  small,  and  great  accuracy  is  required,  the  following 
method  may  be  used  for  sines,  tangents,  and  cotangents. 

I.   Suppose  it  were  required  to  find  the  logarithmic  sine  of  5'  24" 
By  the  ordinary  meth'^i  VQ  should  have 

lo<x.  sin.  5'  =  7.162696 

diff.  for  24"      =       31673 


log.  sin.  5'  24"  --  7.194369 

Ttic  more  accurate  method  is  founded  on  the  proposition  in  Trigo 
nometry,  that  the  sines  or  tangents  of  very  small  angles  are  propor 
tional  to  the  angles  themselves.  In  the  present  case,  therefore,  we 
have  sin.  5':  sin.  5'  24'   =  5'  :  5'  24  '  =  300"  :  324".     Hence  sin.  .5'  24' 

=  '"'  ^.^""'  ,  or  log.  sin.  5'  24"  =  log.  sin.  5'  +  log.  324  —  log.  30ii 

The  difference  for  24"  wiU  therefore,  be  the  difference  between  tlie 
logarithm  of  324  and  the  logarithm  of  SCO.  The  operation  will  stand 
thus  :  — 

log.  324  =  2..510.145 

locr.  300  =2.477121 


diff.  for  24  =       33424 

los.  sin.  5'  =  7.162696 


W.  sin.  5'  24"  =  7.196120 


■'o 


Comparing  this  value  with  that  given  in  tables  that  extend  to  seconds 
we  find  it  exact  even  to  the  last  figure 


PI 


II.   Given  log.  sin.  A  =  7.004438  to  find  A.     The  sine  next  less 
than  this  in  the  table  is  sin.  3  =  6.940817.    Now  we  have  sin.  3' :  sin.  A 

=  3  .  A.     Therefore,  A  =  "1]^7  >  oi"  log-  ^  =  ^^S-  ^  +  ^^o-  ^^^-  ^^ 
-  log.  sin.  3'.     Hence  it  appears,  that,  to  find  the  logarithm  of  A  in 


COSINES,    TANGENTS,    AND    COTANGENTS.  173 

minutes,  we  must  add  to  the  logarithm  of  3  the  difference  oetween 

lojr.  sin,  A  and  log.  sin.  3*. 

log.  sin.  ^1  =  7.004438 

loiT.  sin.  3'  =-  6.940S47 


G3591 
W,  3  =  0.477121 


A  --=  3.473       0.540712 

r,j.  4  ^  3/  28.38".     By  the  common  method  we  should  have  found 

A  =  3'  30. .54". 

The  same  method  applies  to  tangents  and  cotangents,  except  that  in 
the  case  of  cotangents  the  differences  are  to  be  subtracted. 


•  *  The  radius  of  this  table  is  unity,  and  the  characteristics  %  8,  7, 
and  6  stand  respectively  for  —1,  —2,  —3,  and  —4. 


174 

0^ 


TABLE     Xlll,      «f.OGARlTHMIC    SINES, 


179^ 


M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
30 

3r 

33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


Inf.  neg. 

6.463726 
.764756 
.940347 

7.0657S6 
.162696 
.241877 
.303324 
.366S16 
.417963 

7.463726 
.505113 
.542906 
.577663 
.609353 
.639316 
.667345 
.694173 
.718997 
.742478 

7.764754 
.735943 
.806146 
.325451 
.3439.34 
.361662 
.S7S695 
.895035 
.910379 
.926119 

7.940342 
.9.55032 
.963370 
.9S2233 
.995193 

3.007737 
.020021 
.031919 
.043501 
.054731 

8.065776 
.076500 
.036965 
.097133 
.107167 
.116926 
.126471 
.13.5310 
.1449.53 
.15-3907 

8.162631 
.171230 
.179713 
.137935 
.196102 
.204070 
.211395 
.219531 
.2271.34 
.2.34.557 
.241855 

Cosine. 


D.  1  . 


5017.17 

2934.85 

2052.31 

1615.17 

1319.69 

1115.73 

966.53 

8.52. 54 

762.62 

639.33 
629.81 
579.37 
536.41 
499.33 
467.14 
433.31 
413.72 
391.35 
371.27 

353.15 
336.72 
321.75 
303.05 
295.47 
233.33 
27.3.17 
263.23 
253.99 
245.33 

237.33 
229.80 
222.73 
216.03 
209.81 
203.90 
193.31 
193.02 
133.01 
133.25 

173.72 
174.42 
170.31 
166.39 
162.65 
159.03 
155.66 
152.33 
149.24 
146.22 

14.3..33 
140.54 
137.36 
135.29 
132.80 
130.41 
123.10 
125.87 
123.72 
121.64 

D.  1". 


Cosine. 


0.000000 
.000000 
.000000 
.000000 
.000000 
.000000 

9.999999 
999999 
.999999 
.999999 

9.999993 
.999993 
.999997 
.999997 
.999996 
.999996 
.999995 
.999995 
.999994 
.999993 

9.999993 
.999992 
.999991 
.999990 
.999939 
.999939 
.999933 
.999937 
.999936 
.999935 

9.999933 
.999932 
.999931 
.999930 
.999979 
.999977 
.999976 
.999975 
.999973 
.999972 

9.999971 
.999969 
.999963 
.999966 
.999964 
.999963 
.999951 
.999959 
.999953 
.999956 

9.9999.54 
.999952 
.999950 
.999943 
.999946 
.999944 
.999942 
.999940 
.999933 
.999936 
.999934 

Sine. 


D.  1' 


.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.01 
.01 

.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 
.01 

.01 
.01 
.01 
.01 
.02 
.02 
.02 
.02 
.02 
.02 

.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 
.02 

.02 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 

.03 
.03 
.03 
.03 
.03 
.03 
.03 
.04 
.04 
.04 

D.  1". 


Tang. 


D.  1". 


Inf.  neg. 

6.463726 
.764756 
.940347 

7.0657S6 
.162696 
.241373 
.303525 
..366317 
.417970 

7.463727 
.505120 
.542909 
.577672 
.609357 
.639320 
.667849 
.694179 
.719003 
.742484 

7.764761 
.73.5951 
.806155 
.825460 
.843944 
.561674 
.873703 
.895099 
.910394 
.926134 

7.94035S 
.9.55100 
.963589 
.932253 
.99.5219 

S.  007809 
.020044 
.031945 
.043527 
.054309 

8.06^5306 
.07653' 
.036997 
.097217 
.107203 
.116963 
.126510 
.135351 
.144996 
.153952 

8.162727 
.171323 
.179763 
.183036 
.196156 
.204126 
.211953 
.219641 
.227195 
.234621 
.241921 

Cotang. 


5017.17 

29.34.85 

2032.31 

1615.17 

1319.69 

111.5.73 

966.54 

852.55 

762.63 

639.33 
629.81 
-579.37 
536.42 
499.39 
467.15 
433.82 
413.73 
391.36 
371.23 

353.16 
336.73 
321.76 
.303.07 
295.49 
233.90 
273.13 
263.25 
254.01 
245.40 

237.  .35 
229.32 
222.75 
216.10 
209.83 
203.92 
193.33 
193.05 
183.03 
183.27 

173.75 
174.44 
170.a4 
166.42 
162.63 
159.11 
155.69 
1.52.41 
149.27 
146.25 

143.36 
140.57 
137.90 
135.32 
132.84 
130.44 
123.14 
12.5.91 
123.76 
121.63 

D.  1". 


Cotang.  I  M. 


Infinite. 

3.536274 
.235244 
.0.591.53 

2.934214 
.337304 
.753122 
.691175 
.633133 
.5320.30 

2.536273 
.494830 
.457091 
.422323 
.390143 
.360180 
.332151 
.305321 
.230997 
.2.57516 

2.23-5239 
.214049 
.19.3345 
.174540 
.156056 
.133326 
.121292 
.104901 
.039106 
.073366 

2.059142 
.044900 
.031111 
.017747 
.004781 

1.992191 
.979956 
.963055 
.956473 
.945191 

1.934194 
.923469 
.913003 
.9027-33 
.892797 
.83.3037 
.873490 
.664149 
.85.50f>4 
.&46043 

1.837273 

.823672 
.8202.37 
.811964 
.80-3344 
.795874 
.783047 
.780359 
.772305 
.765379 
.753079 

Tang. 


90O 


89" 


COSINES,    TANGENTS,    AND    COTANGENTS. 


175 

1T83 


M. 


Sine 


11 

12 
.3 
14 
15 
.(] 
17 
13 
.9 

20 
?1 
>£;'2 
23 
24 
iij 
28 
27 

29 

30 
32 
32 
33 
34 
35 
36 
3' 
38 
39 

40 
41 
42 
43 
41 
45 
Ifi 
47 
43 

« 

6i 

o2 
53 
54 

55 
56 
57 
58 
59 
60 


D.  1' 


8,2418r)-> 
.213033 
.256' 19-1 
.263012 
.2693S1 
.276614 
.2S3213 
.2Si)773 
.296^07 


8.303794 
.3149.54 
.321027 
.327016 
.332924 
.333753 
.3-14504 
.350131 
.3.J5733 
.361315 

8.366777 
.372171 
.377499 
.332762 
.337962 
.3931111 
.393179 
.4031':!9 
.403161 
.413063 

S.417919 
.422717 
.427462 
.432156 
.436300 
.441.394 
.445941 
.450440 
.454393 
.459301 

8.463665 
,467935 
.472263 
.476493 
.480693 
.4.34343 
.483963 
.493040 
.497073 
.501030 

5.505015 
.503974 
.512^67 
.516726 
.520551 
.524313 
..523102 
.531823 
.535.523 
.539136 
.542319 


Cosine. 


119.63 
117.69 
115.30 
113.93 
11221 
110.50 
103.  S3 
107.22 
105.66 
104.13 

102.66 
101.22 
99.82 
93.47 
97.14 
95.86 
94.60 
93.38 
92.19 
91.03 

89.90 
83.80 
87.72 
86.67 
85.64 
84.64 
83.66 
82.71 
81.77 
80.36 

79.96 
79.09 
78.23 
77.40 
76.53 
75.77 
74.99 
74.22 
73.47 
72.73 

72.00 
71.29 
70.60 
69.91 
69.24 
63.-59 
"67.94 
67.31 
66.69 
66.03 

65.43 
61.89 
64.32 
63.75 
63.19 
62.65 
62.11 
61.53 
61.06 
60.55 


—  I 


M..  I  Cosine. 


D  1'' 


9.999934 
.999932 
.999929 
.999927 
.999925 
.999922 
.999920 
.999913 
.999915 
.999913 

9.999910 
.999907 
.999905 
.999902 
.999399 
.999397 
.999394 
.999391 
.999333 
.999335 

9.999332 
.999379 
.999376 
.999373 
.999370 
.999367 
.999364 
.999361 
.999353 
.999354 

9.999351 
.999348 
.999344 
.999341 
,999333 
.999334 
.999331 
.999327 
.999324 
.999320 

9.999316 
.999313 
.999309 
.999305 
.999301 
.999797 
.999794 
.999790 
.999736 
.999732 

9.999773 
.999774 
.999769 
.999765 
.999761 
.9997.57 
.999753 
.999743 
.99974  4 
.999740 
.9997,35 


D.  1". 


Sine. 


.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 

.04 
.04 
.04 
.05 
.05 
.05 
.05 
.05 
.05 
.05 

.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 

.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 
.06 

.06 
.06 
.06 
.06 
.06 
.06 
.07 
.07 
.07 
.07 

.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 


Tang. 


D.  1". 


D.  1". 


3.241921 
.249102 
.256165 
.263115 
.269956 
.276691 
.283323 
.239356 
.296292 
.302634 

8.303834 
.31.5046 
.321122 
.327114 
.333025 
.3338.56 
.314610 
.350289 
.355895 
.361430 

8.366395 
.372292 
.377622 
.332839 
.338092 
.39.3234 
.398315 
.403333 
.408304 
.413213 

8.418063 
.422369 
.427613 
.432315 
.436962 
.441.560 
.446110 
.450613 
.455070 
.459431 

8.463349 
.463172 
,4724.54 
.476693 
.430-92 
.485050 
.489(70 
.493250 
.497293 
.501293 

8.505267 
.509200 
.513998 
.516961 
.520799 
..524536 
..523349 
.532030 
.535779 
.5.39447 
.543034 


Cotang. 


M. 


119.67 
117.72 
115.84 
114.02 
112.25 
110.54 
103.87 
107.26 
105.70 
104.18 

102.70 
101.26 
99.87 
93.51 
97.19 
95.90 
94.65 
93.43 
92.24 
91.08 

89.95 
88.85 
87.77 
86.72 
85.70 
84.69 
83.71 
82.76 
81.82 
80.9] 

80.02 
79.14 
78.29 
77.45 
76.63 
75.83 
75.05 
74.23 
73.53 
72.79 

72.06 
71.35 
70.66 
69.93 
69.31 
63.65 
63.01 
67.33 
66.76 
66.15 

65. 55 
64.96 
64.39 
63.82 
63.26 
62.72 
62.18 
m.65 
61.13 
60.62 


1.758079 
.750893 
.743<35 
.736335 
,730044 
.723309 
.716677 
.710144 
.703703 
.697366 

1.691116 
,6349.54 
,673373 
672S86 
666975 
.661144 
.65.5390 
.619711 
.644105 
.633570 

1.6.33105 
.627703 
,622373 
.617111 
.611903 
.606766 
.601635 
.596662 
.591696 
.536737 

1.5319.32 
,577131 
.572332 
.567635 
.563033 
.553440 
,553890 
.549337 
.544930 
.540519 

1,5.36151 
.531823 
..527546 
.523307 
.519103 
.514950 
,510330 
.506750 
,.502707 
.493702 

1.494733 
.490300 
.436902 
.483039 
.479210 
.475414 
.471651 
.467920 
.464221 
.460553 
.456916 


CoUin?.  I  D.  1". 


Tang. 


60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
43 
47 
46 
45 
41 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 

29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 


91'^ 


889 


176 

3=" 


TABLE    XIII.       LOGAHITHMIC    SINES, 


173" 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
II 
12 
13 

14 
15 
16 
17 
IS 
19 

20 
21 
22 
23 
21 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
3S 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


8.542319 
.546422 
..549995 
.553.539 
,557054 
,560540 
,563999 
.567431 


.Ol 


4214 

8.577566 
.580392 
,584193 
,587469 
.590721 
,593943 
.597152 
.6J0332 
.603439 
.606623 

8.6097.34 
.612323 
.615391 
.613937 
.621962 
.62^965 
.627943 
.631911 
.6333.54 
.636776 

8.63G630 
.642563 
.64:5423 
.643274 
.651102 
.653911 
.636702 
.659475 
.662230 
.664963 

8.667639 
.670393 
.673030 
.675751 
.678405 
.631043 
.6-3665 
.6-6272 
.633-63 
.6914.33 

8.693998 
.696.543 
.699)73 
.701539 
.704090 
.706.377 
,709049 
,711.507 
.713952 
.716333 
.718300 

Cosine. 


D,  1". 


60.04 
59.55 
59.06 
53.53 
53.11 
57.65 
57.19 
56.74 
56.30 
55.57 

55.44 
55.02 
54.60 
54.19 
53.79 
53.39 
53.00 
52.61 
52.2:3 
51.86 

51.49 
51.12 
50.77 
50.41 
50.06 
49.72 
49.33 
49.04 
48.71 
43.39 

43.06 
47.75 
47.43 
47.12 
46.32 
46.52 
46.22 
4.5.93 
45.63 
45.35 

45.07 
44.79 
44.51 
44.24 
43.97 
43.70 
43.44 
43.18 
42.92 
42.67 

42.42 
42.17 
41.93 
41.63 
41.44 
41.21 
40.97 
40.74 
40.51 
40.29 

D.  1". 


Cosine.   D.  1". 


9.999735 
.999731 
.999726 
.939722 
.999717 
.999713 
.999703 
.999704 
.999699 
.999694 

9.999639 
.999635 
.999630 
.999675 
.999670 
.999665 
.999660 
.9996.55 
.999650 
.999645 

9.999640 
.999635 
.999629 
.999624 
.999619 
.999614 
.999608 
.999603 
.999.597 
.999592 

9.9995S6 
.999581 
.999575 
.999570 
.999564 
.999.553 
.999.553 
.999.547 
.999.541 
.999535 

9.999529 
.999.524 
.999513 
.999512 
.999506 
.999500 
.999493 
.999437 
.999431 
.999475 

9.999469 
.999463 
.999456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999413 
.999411 
.999404 

Sine, 


.07 
.07 
.03 
.03 
.03 
.08 
.08 
,08 
.08 
.08 

,03 
,03 
.03 
.03 
,03 
,03 
.08 
.03 
.03 
.09 

.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 
.09 

.09 
.09 
.09 
.09 
.09 
.10 
,10 
,10 
,10 
,10 

,10 
.10 
.10 
,10 
,10 
.10 
.10 
.10 
,10 
,10 

,10 

,1 

,1 

,1 

,1 

,1 

,1 

,1 

,1 

,1 


D.  1". 


Tang, 


D.  1". 


8.543034 
.546691 
.550268 
.553317 
.557336 
.56J328 
.564291 
..567727 
1137 


.0/ 

.57 


4520 


8.577877 
,531203 
.584514 
.537795 
.591051 
,594283 
,597492 
.600677 
.603339 
.606973 

8.610094 
.613189 
.616262 
.619313 
.622343 
.62-53.52 
.623340 
.631303 
.634256 
.637134 

8.640093 
.6429-2 
.645353 
.643701 
.651.537 
.654352 
.6.57149 
.659923 
.662639 
.66.5433 

8.603160 
.670370 
.673563 
.676239 
.673900 
.631544 
.634172 
.636784 
.639331 
,691963 

8.694529 
.697081 
.699617 
.702139 
.704046 
,707140 
.709618 
.712033 
.714534 
.716972 
.719396 

Cotang. 


60.12 
59.62 
59.14 
58.66 
58.19 
57.73 
57.27 
56.62 
56.38 
55.95 


5.5.10 
54.63 
54.27 
53.87 
53.47 
53.08 
52.70 
52.32 
51.94 

51.58 
51.21 
50.85 
50.50 
50.15 
49.81 
49.47 
49.13 
48.80 
.  48.48 

48.16 
47.84 
47.  .53 
47.22 
46.91 
46.61 
46.31 
46.02 
45.73 
45.45 

45.16 
44.33 
44.61 
44.34 
44.07 
4.3.30 
43.54 
43.23 
43.03 
42.77 

42.52 
42.23 
42.03 
41.79 
41.55 
41.32 
41.08 
40.85 
40.62 
40.40 

D,  1". 


Cotang. 

M. 

1.456916 

60 

.453309 

59 

.4497.32 

58 

.446183 

.57 

.442664 

.56 

,439172 

55 

.435709 

54 

,432273 

53 

.428863 

52 

,425480 

51 

1.422123 

50 

.418792 

49 

,41.5436 

48 

,412205 

47 

,408949 

46 

,405717 

45 

.402503 

44 

,399323 

43 

.396161 

42 

,39.3022 

41 

1.389906 

40 

,3.36311 

39 

.383738 

38 

330637 

37 

377657 

36 

,374648 
.371660 
.36-692 
.365744 
.362816 

1.359907 
.357018 
,3.54147 
..351296 
.343463 
.345643 
.342851 
..340072 
.3.37311 
.a34567 

1.331840 
.3291.30 
.32&4.37 
•323761 
.321100 
.313456 
.315323 
.313216 
.310619 
.308037 

1.30.5471 
.302919 
.300383 
.297861 
,295354 
,292360 
.290382 
,287917 
.28.5466 
.253023 
,280604 

Tang. 


9«3 


•il-' 


COSINES,    TAiMGENTS,    AND    COTAKGENTS. 


n7 

176^ 


M 

0 
1 
2 
3 

4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
83 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine 


D  1". 


Cosine. 


D.  1" 


. '-.718300 
.7212)4 
.723595 
.725972 
.723337 
.730633 
.733027 
.735354 
.737667 
.739969 

S.  742259 
.744536 
.746302 
.749055 
.751297 
.753523 
.755747 
.7579")5 
.76)151 
.762337 

3.761511 
.766675 
.76S323 
.770970 
.773101 
.77.5223 
.777333 
.779434 
.781524 
.733605 

3.735675 
.737736 
.739737 
.791323 
.793359 
.795331 
.797394 
.799397 
.801392 
.803376 

8.80.5852 
.807819 
.809777 
.311726 
.81.3657 
.81.5.599 
.317522 
.819436 
.321313 
.323240 

8.325130 
.327011 
.823384 
.8.30749 
.832607 
.834456 
.836297 
.8.33130 
.339956 
.341774 
.843535 


40.06 
39. -^4 
39.62 
39.41 
39.19 
33.93 
33.77 
33.57 
33.36 
33.16 

37.96 
37.76 
37.56 
37.37 
37.17 
36.93 
36.30 
36.61 
36.42 
36.24 

36.06 
35.83 
35.70 
35.53 
35.. 35 
35.13 
35.01 
31.31 
34.67 
31.51 

31.31 
31.18 
34.02 
.33.36 
33.70 
33.54 
33.39 
.33.23 
33.03 
32.93 

32.73 
32.63 
32.49 
32.34 
32.20 
32.05 
31.91 
31.77 
31.63 
31.49 

31.36 
31.22 
31.03 
30.95 
30.82 
30.69 
30.56 
30.43 
30.30 
30.17 


9.999404 
.999393 
.999391 
.999334 
.999378 
.999371 
.999364 
.999357 
.9993.50 
.999313 

9.999336 
.999329 
.999.322 
.999315 
.999303 
.999301 
.999294 
.999237 
.999279 
.999272 

9.999265 
.999257 
.9992.50 
.999242 
.999235 
.999227 
.999220 
.999212 
.999205 
.999197 

9.999189 
.999181 
.999174 
.999166 
.999153 
.999150 
.999142 
.999131 
.999126 
.999113 

9.999110 
.999102 
.999094 
.999036 
.999077 
.999069 
.999':)61 
.999053 
.999044 
.999036 

9.999027 
.999019 
.999010 
.999002 
.993993 
.993931 
.993976 
.993967 
.993953 
.9939.50 
.99-^941 


Tang. 


D.  1" 


Cosine.  I  D.  1". 


Sine 


.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.12 
.12 

.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 
.12 

.12 
.12 
.12 
.12 
.13 
.13 
.13 
.13 
.13 
.13 

.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

.14 
.11 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 

.14 
.14 
.14 
.14 
.14 
.14 
.15 
.15 
.15 
.15 


D.  1". 


8. 71 9396 
.721306 
.724204 
.726533 
.723959 
.731317 
.733663 
.735996 
.733317 
.740626 

8.742922 
.745207 
.747479 
.749740 
.751939 
.754227 
.756453 
.753663 
.760372 
.763065 

8.765216 
.767417 
.769573 
.771727 
.773366 
.775995 
.773114 
.730222 
.732320 
.784403 

8.736436 
.7335.54 
.790613 
.792662 
.794701 
.796731 
.793752 
.800763 
.802765 
.804753 

8.806742 
.803717 
.810633 
.812641 
.3145^9 
.816529 
.813161 
.320334 
.822293 
.824205 

3.326103 
.827992 
.829374 
.831743 
.833613 
.83^5471 
.837.321 
.839163 
.840993 
.842325 
.844644 


Cotang. 


Cotang. 


40.17 
39.95 
39.74 
39.52 
39.31 
39.10 
33.89 
33.63 
33.43 
38.27 

38.07 
37.83 
37.63 
37.49 
37.29 
37.10 
36.92 
36.73 
36.55 
36.36 

36.18 
36.00 
35.83 
35.65 
35.43 
35.31 
35.14 
31.97 
34.80 
34.64 

34.47 
34.31 
34.15 
33.99 
33.83 
33.63 
33.52 
33.37 
33.22 
33.07 

32.92 
32.77 
32.62 
.32.43 
32.33 
32.19 
32.05 
31.91 
31.77 
31.63 

31. .50 
31.36 
31.23 
31.09 
30.96 
30.83 
30.70 
30.57 
30.45 
30.32 


D.  1". 


1.230604 
.273194 
.275796 
.273412 
.271011 
.263633 
.266337 
.264004 
.261633 
.259374 

1.257078 
.254793 
.252521 
.250260 
.243011 
.245773 
.243.547 
.241332 
.239123 
.236935 

1.234754 
.232533 
.230422 
.223273 
.226131 
.224005 
.221886 
.219773 
.2176i0 
.215592 

1.213514 
.211446 
.209337 
.207333 
.205299 
.203269 
.201243 
.199237 
.1972.35 
.195212 

1.193253 
.191233 
.189317 
.137359 
.135411 
.133471 
.131.539 
.179616 
.177702 
.17.5795 

1.173397 
.172003 
,170126 
.163252 
,166337 
.164529 
.162679 
.1603.37 
.159002 
.157175 
.1.55356 


Tang. 


M. 

60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 

23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 


86<? 


178 

40 


TABLE    XIII.       LOGARITHMIC    SINES, 


175' 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
il 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24- 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine. 


8.S435S5 
.S453S7 
.847183 
.348971 
.850751 
.852.525 
.8-54291 
.856049 
.857801 
.859546 

8.S612S3 
.863014 
.864733 
.866455 
.863165 
.S69S63 
.871.565 
.87.3255 
.874933 
.876615 

8.878235 
.879949 
.831607 
.333253 
.834903 
.836542 
.838174 
.889301 
89 1 421 
.893035 

8.394643 
.896246 
.897342 
.899432 
.901017 
.902596 
.904169 
.905736 
.907297 
.903353 

8.910404 
.911949 
.913433 
.91.5022 
.916550 
.918073 
.919'.91 
.921103 
.922610 
.924112 

8.925609 
.927100 
.923587 
.930063 
.931544 
.933015 
.934431 
.935942 
.937393 
.938350 
.940296 


1  M.      Cosine. 


^o 


D.  1". 


30.05 
29.92 
29.80 
29.63 
29.-55 
29.43 
29.31 
29.19 
29.03 
28.96 

23.84 
28.73 
23.61 
23.50 
23.39 
28.23 
23.17 
23.06 
27.95 
27.34 

27.73 
27.63 
27.52 
27.42 
27.31 
27.21 
27.11 
27.00 
26.90 
26.30 

26.70 
26.60 
26.51 
26.41 
26.31 
26.22 
26.12 
26.03 
25.93 
25.34 

25.75 
25.66 
25.56 
25.47 
25.38 
25.29 
25.21 
25.12 
25.03 
24.94 

24.86 
24.77 
24.69 
24.60 
24.52 
24.43 
24.-35 
24.27 
24.19 
24.11 


Cosine. 


D.  1'. 


9.99S94I 
.993932 
.993923 
.993914 
.993905 
.993896 
.998.387 
.993378 
.993369 
.993360 

9.993351 
.993341 
.993332 
.993323 
.993313 
.993304 
.993795 
.993785 
.993776 
.993766 

9.998757 
.993747 
.998733 
.993723 
.993718 
.993703 
.993699 
.993639 
.995679 
.993669 

9.998659 
.993649 
.993639 
.993629 
.993619 
.99S609 
.993599 
.993.589 
.993573 
.993563 

9.993553 
.993-S43 
.993537 
.993527 
.993516 
.993506 
.993495 
.993485 
.998474 
.993464 

9.993453 
.993442 
.993431 
.993421 
.993410 
.993399 
.993338 
.993377 
.993.366 
.9933-55 
.993344 


D.  1". 


Sine. 


.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 

.15 
.15 
.15 
.16 
.16 
.16 
.16 
.16 
.16 
.16 

.16 
.16 
.16 
.16 
.16 
.16 
.16 
.16 
.16 
.17 

.17 
.17 
.17 
.17 
.17 
.17 
.17 
.17 
.17 
.17 

.17 
.17 
.17 
.17 
.17 
.18 
.18 
.18 
.18 
.18 

.18 

18 

.18 

18 
.18 
.18 
.18 
.18 
.18 
.18 


Tang. 


D.  1". 


8.844644 
.346455 
.843260 
.850057 
.851846 
.853628 
.85.5403 
.857171 
.858932 
.860636 

8.S62433 
.864173 
.865906 
.867632 
.869-351 
.871064 
.872770 
.874469 
.876162 
.877349 

8.879529 
.881202 
.832369 
.834530 
.886185 
.837833 
.889476 
.891112 
.892742 
.894366 

8.895934 
.897596 
.899203 
.900303 
.902398 
.90-3937 
.905570 
.907147 
.903719 
.910235 

8.911346 
.913401 
.914951 
.916495 
.9130-34 
.919563 
.921096 
.922619 
.924136 
.925649 

8.927156 
.923658 
.9-301-55 
.931647 
.9-33134 
.934616 
.936093 
.937565 
.9390-32 
.940494 
.941952 


D  1".       Cotang. 


30.20 
30.07 
29.95 
29.83 
29.70 
29.53 
29.46 
29.35 
29.23 
29.11 

29.00 
23.88 
23.77 
23.66 
23.55 
23.43 
23.32 
23.22 
23.11 
23.00 

27.89 
27.79 
27.63 
27.-58 
27.47 
27.37 
27.27 
27.17 
27.07 
26.97 

26.87 
26.77 
26.67 
26.58 
26.48 
26.-39 
26.29 
26.20 
26.10 
26.01 

2-5.92 
25.83 
25.74 
25.65 
25.56 
25.47 
25.38 
25.29 
25.21 
25.12 

25.04 
24.95 
24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.29 


Cotang.  1  D.  1". 


1.1553S6 
.153545 
.151740 
.149943 
.143154 
.146372 
.144597 
.142329 
.141063 
.139314 

1.137567 
.135327 
.1-34094 
.132363 
.130649 

•  .123936 
.127230 
.125531 
.123333 
.122151 

1.120471 

.113793 
.117131 
.11.5470 
.113315 
.112167 
.110524 
.108383 
.107258 
.105634 

1.104016 
.102404 
.100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091231 
.039715 

LOSS  154 
.086599 
.085049 
.033505 
.081966 
.080432 
.078904 
.077381 
.075864 
.074351 

1.072344 
.07i:J42 
.069345 
.0633.53 
.066366 
.065384 
.063907 
.062435 
.060968 
.059506 
.058048 


Tang. 


8»<3 


COSINES,    lANGENTS,    AND    COTANGENTS. 


M. 


Sine. 


D.  1". 


0 

8.940296 

1 

.941733 

2 

.913174 

3 

.944606 

4 

.946034 

5 

.947456 

6 

.948374 

7 

.950237 

8 

.951696 

9 

.953100 

10 

3.954499 

11 

.955394 

12 

.957234 

13 

.953670 

14 

.960052 

15 

.961429 

16 

.962301 

17 

.964170 

18 

.965531 

19 

.933993 

20 

3.963249 

21 

.969600 

22 

.970947 

23 

.972239 

24 

.973623 

25 

.974902 

26 

.976293 

27 

.977619 

23 

.978941 

.930259 

3l 

^  8.931573 

31 

.932333 

32 

.934189 

33 

.93.5491 

34 

.936789 

35 

.933033 

36 

.939374 

37 

.990660 

33 

.991943 

39 

.993222 

40 

3.994497 

41 

.995763 

42 

.997036 

43 

.993299 

44 

.999560 

45 

9.000316 

46 

.002069 

47 

.003318 

43 

.004563 

49 

.005305 

50 

9.007044 

51 

.003278 

52 

.009510 

53 

.010737 

54 

.011962 

55 

.013182 

56 

.014400 

57 

.015613 

53 

.016324 

59 

.018031 

60 

.019235 

M. 


»5<3 


Cosine. 


24.03 
23.95 
23.87 
23.79 
23.71 
23.63 
23.55 
23.48 
23.40 
23.32 

23.25 
23.17 
23.10 
23.02 
22.95 
22.83 
22.31 
22.73 
22.66 
22.59 

22.52 
22.45 
22.33 
22.31 
22.24 
22.17 
22.10 
22.03 
21.97 
21.90 

21.33 
21.77 
21.70 
21.64 
21. .57 
21.51 
21.44 
21.33 
21.31 
21.25 

21.19 
21.12 
21.06 
21.00 
20.94 
20.83 
20.82 
20.76 
20.70 
20.64 

20.53 
20.52 
20.46 
20.40 
20.35 
20.29 
20.23 
20.17 
20.12 
20.06 


D.  1". 


9.99S344 
.993333 
.993322 
.998311 
.993300 
.998239 
.993277 
.993266 
.993255 
.993243 

9.993232 
.998220 
.993209 
.993197 
.993186 
.993174 
.993163 
.998151 
.993139 
.993123 

9.99SI16 
.993104 
.993092 
.993030 
.993063 
.993056 
.993044 
.993032 
.993020 
.993008 

9.997996 
.997984 
.997972 
.997959 
.997947 
.997935 
.997922 
.997910 
.997897 
.997885 

9.997372 
.997360 
.997847 
.997335 
.997822 
.997809 
.997797 
.997734 
.997771 
.997753 

9.997745 
.997732 
.997719 
.997706 
.997693 
.997630 
.997667 
.9976;j-l 
.997641 
.997623 
.997614 


Cosine.   D.  1". 


Sine. 


.18 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 
.19 

.19 
.19 
.19 
.19 
.19 
.19 
.19 
.20 
.20 
.20 

.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 

.20 
.20 
.20 
.20 
.21 
.21 
.21 
.21 
.21 
.21 

21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 
.21 

.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 
.22 


Tans. 


8.941952 
.94:^0-1 
.944352 
.946295 
.947734 
.949168 
.950597 
.952021 
.953441 
.954356 

8.956267 
.957674 
.959075 
.960473 
.961866 
.963255 
.964639 
.966019 
.967394 
.963766 

8.970133 
.971496 
.972355 
.974209 
.975560 
.976906 
.978243 
.979536 
.930921 
.932251 

8.933577 
.934899 
.936217 
.937532 
.938342 
.990149 
.991451 
.992750 
.994045 
.995337 

8.996624 
.997903 
.999188 

9.000465 
.001733 
.0^3007 
.004272 
.005534 
.006792 
.008047 

9.009293 
.010546 
.011790 
.013031 
.014268 
.015502 
.016732 
.017959 
.019183 
.020403 
.021620 


D.  1". 


D.  1'. 

24.21 
24.13 
24.05 
23.97 
23.90 
23.82 
23.74 
23.67 
23.59 
23.51 

23.44 
23.36 
23.29 
23.22 
23.14 
23.07 
23.00 
22.93 
22.86 
22.79 

22.72 
22.65 
22.58 
22.51 
22.44 
22.37 
22.30 
22.24 
22.17 
22.10 

22.04 
21.97 
21.91 
21.84 
21.78 
21.71 
21.65 
21.59 
21.52 
21.46 

21.40 
21.34 
21.27 
21.21 
21.15 
21.09 
21.03 
20.97 
20.91 
20.85 

20.80 
20.74 
20.63 
20.62 
20.56 
20.51 
20.45 
20.39 
20.34 
20.28 


Cotang. 


179 

174-0 

M. 


Cotang. 


1.058048 
.056596 
.055148 
.053705 
.052266 
.050332 
.049403 
.047979 
.046559 
.045144 

1.0437.33 
.042326 
.040925 
.039527 
.038134 
.036745 
.035361 
.033981 
.032606 
.031234 

1.029367 
.023504 
.027145 
.025791 
.024440 
.023094 
.021752 
.020414 
.019079 
.017749 

1.016423 
.015101 
.013783 
.012468 
.011153 
.009851 
.008549 
.007250 
.005955 
.004663 

1.003376 
.002092 
.000312 

0.999535 
.993262 
.996993 
.995728 
.994466 
.993203 
.991953 

0.990702 
.939454 
.988210 
.986969 
.985732 
.984498 
.933268 
.982041 
.980817 
.979597 
.978380 


D.  1". 


6C 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


Tang.   M- 


84<3 


180 

60 


C-  0    i    y  ^  <? 
TABLE    Xlll.       LOGARITHMIC    SINES, 


173^ 


M. 

0 
1 
•2 
3 
4 


Sine. 


D.  1' 


10 
11 
12 
13 
14 
15 
16 
17 
IS 
19 

20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.019235 
.020435 
.021632 
.0228-25 
.024016 
.02.5203 
.026366 
.027567 
.028744 
.029918 

9.0310-9 
.032257 
.033421 
.034582 
.0.35741 
.036896 
.038048 
.039197 
.040.342 
.041485 

9.042625 
.013762 
.044895 
.046026 
.047154 
.048279 
.049400 
.050519 
.051635 
.0.52749 

9.053859 
.054966 
.056071 
.057172 
.058271 
.059367 
.060460 
.061.551 
.062639 
.063724 

9.064-06 
.065885 
.066962 
.06-036 
.069107 
.070176 
.071242 
.0723% 
.07.3366 
.074424 

9.075460 
.076533 
.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 
.084864 
.085894 


M.  !    Cosine. 


20.00 
19.95 
19.89 
19.84 
19.78 
19.73 
19.67 
19.62 
19.57 
19.51 

19.46 
19.41 
19.36 
19.30 
19.25 
19.20 
19.15 
19.10 
19.05 
19.00 

18.95 
18.90 
18.85 
18.80 
18.75 
18.70 
18.65 
18.60 
18.. 55 
18.50 

18.46 
18.41 
18.36 
18.31 
18.27 
18.22 
18.17 
18.13 
18.08 
18.04 

17.99 
17.95 
17.90 
17.66 
17.81 
17.77 
17.72 
17.68 
17.64 
17.59 

17.  .55 
17.51 
17.46 
17.42 
17.38 
17.34 
17.29 
17.25 
17.21 
17.17 


Cosine. 


D.  1". 


9.997614 
.997601 
.997588 
.997574 
.997.561 
.997.547 
.997534 
.997520 
.997507 
.997493 

9.997480 
.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997333 
.997369 
.997355 

9.997341 
.997327 
.997313 
.997299 
.997285 
.997271 
.997257 
.997242 
.997228 
.997214 

9.997199 
.997185 
.997170 
.997158 
.997141 
.997127 
.997112 
.997098 
.997083 
.997063 

9.997053 
.997039 
.997024 
.997009 
.996994 
.996979 
.996964 
.996949 
.996934 
.996919 

9.996904 
.996889 
.996874 
.996858 
.996843 
.996828 
.996812 
.996797 
.996782 
.996766 
.996751 


Sin*. 


D.  1". 

.22 
22 
22 

.22 

,22 
22 

.23 
23 
23 
23 

23 
,23 
.23 
23 
.23 
.23 
.23 
.23 
.23 
.23 

.23 
.23 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.24 

.24 
.24 
.24 
.24 
.24 
.24 
.24 
.24 
.25 
.25 

.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 

.25 
.25 
.25 
.25 
.26 
.26 
.26 
.26 
.26 
.26 


Tang. 


D.  1". 


D.l 


9,021620 
.022834 
.024044 
.025251 
.026455 
.027655 
.028852 
.030046 
.031237 
.032425 

9.033609 
.034791 
.r,35C69 
.337144 
.038316 
.039485 
.040651 
.M1813 
.042973 
.044130 

9.0452^4 
.046134 
.047.582 
.04S727 
.049869 
.051008 
.0.52144 
.053277 
.054407 
.055535 

9.056659 
.0.57781 
.058900 
.060016 
.0611.30 
.062240 
.063.348 
.064453 
.065556 
.066655 

9.067752 
.068846 
.069933 
.071027 
.072113 
.073197 
.074278 
.075.356 
.076432 
.077505 

9.078576 
.079644 
.080710 
.081773 
.082S33 
.08.3^91 
.084947 
.086000 
.087050 
.088098 
.089144 


20.23 
20.17 
20.12 
20.06 
20.01 
19.95 
19.90 
19.85 
19.79 
19.74 

19.69 
19.64 
19.. 58 
19.53 
19.48 
19.43 
19.38 
19.33 
19.28 
19.23 

19.18 
19.13 
19.08 
19.03 
18.98 
18.93 
18.89 
18.84 
18.79 
18.74 

18.70 
18.65 
18.60 
18.56 
18.51 
18.46 
18.42 
IS.. 37 
18.33 
18.28 

18.24 
18.19 
18.15 
18.10 
1S.06 
18.02 
17.97 
17.93 
17.89 
17.84 

17.80 
17.76 
17.72 
17.67 
17.63 
17.59 
17.55 
17.51 
17.47 
17.43 


Cotang  I  D.  1". 


Cotang. 

60 

0.976380 

.977166 

59 

.9759.56 

58 

.974749 

57 

.973545 

56 

.972315 

55 

.971148 

54 

.£699.54 

53 

.968763 

52 

.967575 

51 

0.966391 

50 

.965209 

49 

.964031 

4L 

.962856 

47 

.961684 

46 

.960515 

45 

.9.59349 

14 

.9.58187 

43 

.957027 

42 

.955870 

41 

0.9.54716 

40 

.95.3566 

39 

.952418 

38 

.951273 

37 

.950131 

£6 

.948992 

35 

.947856 

34 

.946723 

33 

.945593 

,32 

.944465 

31 

0.943341 

l'.0 

.942219 

29 

.941100 

28 

.939984 

27 

.938870 

26 

.937760 

25 

.936652 

24 

.935547 

25 

.934441 

22 

.933345 

21 

0.9.32248 

20 

.9311.54 

19 

.930(62 

18 

.928973 

17 

.927887 

16 

.926303 

15 

.925722 

14 

.924644 

13 

.92.3.-68 

12 

.922495 

11 

0.921424 

10 

.920356 

9 

.919290 

8 

.918227 

7 

.917167 

6 

.916109 

5 

.915053 

4 

.914000 

3 

.912950 

2 

.911902 

1 

.S10^C6 

0 
M 

Tang. 

oeo 


830- 


COSINES,    TANGENTS,    AND    COTANGENTS 


181 


M. 

0 
1 
2 
3 

4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 

2;5 

26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine. 


D.  1". 


M. 


9.085394 
.036922 
.037947 
.033970 
.039990 
.091008 
.092024 
.093037 
.094047 
.095056 

9.096062 
.097(165 
.09S066 
.099065 
.100[)62 
.101056 
.102013 
.103037 
.104025 
.105010 

9.105992 
.106973 
.107951 
.105927 
.109901 
.110373 
.111842 
.112309 
.113774 
.114737 

9.115693 

.116656 
.117613 
.113567 
.119519 
,120469 
.121417 
.122362 
.123306 
.124243 

9.125187 
.126125 
.127060 
.127993 
.123925 
.129354 
.130731 
.131706 
.1.32630 
.133551 

9.134470 
.13.53S7 
.136303 
.137216 
.133123 
.139037 
.139944 
.140350 
.141754 
142655 
.143555 

CoBine. 


17.13 
17.09 
17.05 
17.00 
16.96 
16.92 
16.88 
16.84 
16.30 
16.76 

16.73 
16.69 
16.65 
16.61 
16.57 
16.53 
16.49 
16.46 
16.42 
16.33 

16.34 
16.30 
16.27 
16.23 
16.19 
16.16 
16.12 
16.03 
16.05 
16.01 

15.98 
15.94 
1.5.90 

15.87 
15.83 
15.80 
15.76 
15.73 
15.69 
15.66 

15.62 
15.59 
15.56 
15.52 
15.49 
15.45 
15.42 
15.39 
15.35 
15.32 

15.29 
15.26 
15.22 
15.19 
15.16 
15.13 
15.09 
15.06 
15.03 
15.00 

D.  1". 


9.996751 
.996735 
.996720 
.996704 
.9966SS 
.996673 
.996657 
.996641 
.996625 
.996610 

9.996594 
.996573 
.996562 
.996.M6 
.996530 
.996514 
.996198 
.996482 
.996465 
.996449 

9.996433 
.996417 
.996400 
.996334 
.996363 
.996351 
996335 
.996318 
.996302 
.996235 

9.996269 
.996252 
.996235 
.996219 
.996202 
.996135 
.996163 
.996151 
.996134 
.996117 

9.996100 
.996033 
.996066 
.996049 
.996032 
.996015 
.995993 
.995930 
.995963 
.995946 

9.995928 
.995911 
.995394 
.995876 
.995359 
.995341 
.995323 
.995306 
.995783 
.995771 
.995753 

Sine. 


.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 
.26 

.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 
.27 

.27 
.27 
.27 
.27 
.27 
.27 
.23 
.28 
.28 
.23 

.28 
.28 
.28 
.28 
.28 
.23 
.23 
.23 
.23 
.23 

.23 
.23 
.28 
.29 
.29 
.29 
.29 
.29 
.29 
.29 

.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.29 
.30 


9.039144 
.090137 
.091223 
.1)92266 
.093302 
.094336 
.095367 
,096395 
.097422 
.093446 

9.099463 
.100487 
•101504 
.102519 
.103532 
.104542 
.105550 
.1065.56 
.107559 
.108560 

9.109559 
.1105.56 
.111551 
.112543 
.113533 
.114.521 
.115507 
.116491 
.117472 
.118452 

9.119429 
.120404 
.121377 
.122343 
.123317 
.124234 
.125249 
.126211 
.127172 
.123130 

9.129037 
.130041 
.130994 
.131944 
.132393 
.133339 
.1347^34 
.135726 
.136667 
.137605 

9.133542 
.139476 
.140409 
.141310 
.142269 
.143196 
.144121 
.14.5044 
.145966 
.146335 
.147803 


D.  1", 


17.39 
17.35 
17.31 

17.27 
17.23 
17.19 
17.15 
17.11 
17.07 
17.03 

16.99 
16.95 
16.91 
16.83 
16.84 
16.80 
16.76 
16.72 
16.69 
16.65 

16.61 

16.53 

16..54 

16.50 

16.47 

16.43 

16.39 

16.36 

16.32 

16.29 

16.25 
16.22 
16.18 
16.15 
16.11 
16.03 
16.04 
16.01 
15.93 
15.94 

15.91 
15.87 
15.84 
15.81 
15.77 
15.74 
15.71 
15.63 
1.5.64 
15.61 

15.58 
15.55 
15.51 
1.5.43 
15.45 
15.42 
15.39 
1.5.36 
15.32 
15.29 


0.910S56 
.909S13 
.908772 
.907734 
.906693 
.905664 
.904633 
.903605 
.902573 
.901554 

0.900532 
.899513 
.893496 
.897451 
.896463 
.895453 
.894450 
.893444 
.892441 
.891440 

0.890441 

.839444 
.833449 
.837457 
.836467 
,835479 
.884493 
.883509 
.832.523 
,831543 

0.880571 
.879596 
.878623 
.877652 
.876633 
.875716 
.874751 
.873789 
.872328 
.871870 

0.870913 
.869959 
.869006 
.863056 
.867107 
.866161 
.865216 
.864274 
.863333 
.862395 


Cotang. 


D.  1'. 


0.861453 
.860.524 
.859591 
.853660 
.857731 
.856304 
.85.5379 
,854956 
.854034 
,853115 
.852197 


60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 

33 

37  I 

36  I 

35 

34 

33 

32 

31 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


Tang. 


M. 


07- 


8«- 


182 

so 


TABLE    XIII.       LOGARITHMIC    SINES, 


171<i 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 

12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine 


9.143555 
.144453 
.145349 
.146243 
.147136 
.143026 
.143915 
.149302 
.150636 
.151569 

9.152451 
.1.53.330 
.1.34203 
.1.55033 
.1-559.57 
.  1.56330 
.1.57700 
.153569 
.159435 
.160301 

9.161164 
.162025 
.162335 
.163743 
.164600 
.1654.54 
.166307 
.1671.59 
163003 
.1633.56 

9.169702 
.170.547 
.171.339 
.172230 
.173070 
.173903 
.174744 
.175573 
.176411 
.177242 

9.173072 
.173900 
.179726 

..130551 
.181374 
.132196 
.13.3016 
.133831 
.184651 
.135466 

9.136230 
.187092 
.187903 
.183712 
.139519 
.190-325 
.191130 
.191933 
.192734 
.193534 
.194332 

Cosine. 


D.  1". 


4.97 
4.93 
4.90 
4.S7 
4.34 
4.31 
4.78 
4.75 
4.72 
4.69 

4.66 
4.63 
4.60 
4.57 
4-54 
4.51 
4.43 
4.45 
4.42 
4.39 


4.. 36 
4.33 
4.. 30 
4.27 
4.24 
4.22 
4.19 
4.16 
4.13 
4.10 

4.07 
4.05 
4.02 
3.99 
3.96 
3.94 
3.91 
13.83 
-3.35 
3.  S3 

3.80 
3.77 
3.75 
3.72 
3.69 
.3.67 
3.64 
3.61 
3.-59 
3.56 

3.54 
3.51 
3.48 
3.46 
3.43 
3.41 
3.33 
.3.36 
-3.33 
3.31 


D.  1". 


Cosine. 


9.995753 
.995735 
.995717 
.99.5699 
995631 
.995664 
.995646 
.995628 
.995610 
.995591 

9.995573 
.99.5555 
.995537 
.995519 
.995501 
.995432 
.995464 
.995446 
.995427 
.995409 

9.99-5390 
.995372 
.995353 
.995334 
.995316 
.995297 
.995273 
.99.3260 
.995241 
.995222 

9.995203 
.995184 
.995165 
.995148 
.995127 
.995103 
.995039 
.995070 
.995051 
.995032 

9.99.5013 
.994993 
.994974 
.994955 
.994935 
.994916 
.994396 
.994377 
.994357 
.994333 

9.994818 
.994793 
.994779 
.994759 
.994739 
.994720 
994700 
.994630 
.994660 
.994640 
■994620 

Sine. 


D.r 


.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 
.30 

.30 
.30 
.30 
.30 
.30 
.31 
.31 
.31 
.31 
.31 

.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 
.31 

.31 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
..32 
.32 

.32 
.32 
.32 
.32 

.32 
.32 

.a3 

..33 
.33 
.33 

.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 
.33 

D.  1". 


Tang.    D.  1".   Cotang. 


9.147803 
.143713 
.149632 
.150544 
.151454 
.152363 
.153269 
.1.54174 
.15.5077 
.155978 

9.156877 
.157775 
.158671 
.159565 
.1604.57 
.161.317 
.162236 
.163123 
.164003 
.164892 

9.16.5774 
.166654 
.167532 
.168409 
.169234 
.170157 
.171029 
.171899 
.172767 
.173634 

9.174499 
.17.5362 
.176224 
.177084 
.177942 
.178799 
.179655 
.180508 
.181360 
.182211 

9.183059 
.18.3907 
.184752 
,185597 
.186439 
.187280 
.188120 
.188953 
.139794 
.190629 

9.191462 
.192294 
.193124 
.19.3953 
.194730 
.19.5606 
.196430 
.1972.53 
.193074 
.19S394 
199713 

Cuiang. 


15.26 
15.23 
15.20 
15.17 
15.14 
15.11 
1.5.03 
15.05 
15.02 
14.99 

14.96 
14.93 
14.90 
14.87 
14.84 
14.81 
14.78 
14.75 
14.73 
14.70 

14.67 
14.64 
14.61 
14..53 
14.56 
14.-53 
14.50 
14.47 
14.44 
14.42 

14.39 
14.36 
14.33 
14.31 
14.28 
14.25 
14.23 
14.20 
14.17 
14.15 

14.12 
14.09 
14.07 
14.04 
14.02 
1.3.99 
13.97 
13.94 
13.91 
13.89 

13.86 
13.84 
1-3.81 
13-79 
1-3.76 
1-3.74 
13.71 
1-3.69 
13.66 
13.64 

D.  1". 


0.852197 

60 

.851282 

59 

.850363 

58 

.849456 

57 

.848546 

56 

.847637 

55 

.846731 

54 

.845826 

53 

.844923 

52 

.844022 

51 

0.843123 

50 

.842225 

49 

.841329 

43 

.8404.35 

47 

.839543 

46 

.833653 

45 

.837764 

44 

.836377 

43 

.835992 

42 

.835103 

41 

0.834226 

40 

.8-33.346 

39 

S32463 

38 

.831591 

37 

.830716 

36 

.829343 

35 

.828971 

34 

.828101 

33 

.827233 

32 

.826366 

31 

0.825501 

30 

.824633 

29 

.823776 

28 

.822916 

27 

.822053 

26 

.821201 

25 

.820345 

24 

.819492 

23 

.818640 

22 

.817789 

2i 

0.816941 

20 

.816093 

19 

.81.5243 

18 

.814403 

17 

.813561 

16 

.812720 

15 

.811330 

14 

.811042 

13 

.810206 

12 

.809371 

11 

0.808538 

10 

.807706 

9 

.806376 

8 

.806047 

7 

.805220 

6 

.804394 

5 

.803570 

4 

.802747 

3 

.801926 

2 

.801106 

1 

.800237 

0 

Tang. 

M. 

as*  2 


81 


COSINES,    TANGENTS,    AND    COTANGENTS. 


183 

170^ 


0 
1 
2 
3 
4 
5 
6 

i 

S 
9 

10 
11 

12 
13 
14 
15 
16 
17 
IS 
19 

20 
21 
22 
23 
24 
2.", 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
3S 
39 


40 

41 

42 

43 

44 

45 

4(r 

47 

48 

49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.194332 
.195129 
.195925 
.196719 
.197511 
.193.302 
.199091 
.199379 
.200666 
.201451 

9.202234 

.203017 
.203797 
.201577 
.20.5351 
.206 1 31 
.2)6906 
.207679 
.2,03452 
.209222 

9.200992 
.210760 
.211.526 
.212291 
.2131.55 
.213313 
.214579 
.215333 
.216097 
.216354 

9.217609 
.213363 
.219116 
.219363 
.220613 
.221367 
.222115 
.222361 
.223396 
.221349 


9.225092 
.225333 
.226573 
.227311 
.223013 
.223734 
.229513 
.230252 
.230934 
.231715 

9.232114 
.233172 
.233399 
.231625 
.235319 
.236073 
,236795 
.2.37515 
.2332.-5 
.233953 
.2.396"1 


13.28 
13.26 
13.23 
13.21 
13.18 
13.16 
13.13 
13.11 
13.03 
13.  16 

13.MI 
13.01 
12.99 
12.96 
12.91 
12.92 
12.89 
12.37 
12.85 
12.82 

12.80 
12.73 
12.75 
12.73 
12.71 
12.63 
12.66 
12.64 
12.62 
12.59 

12.. 57 
12.55 
12.53 
12.  .50 
12.43 
12.46 
12.44 
12.42 
12.39 
12.37 

12..35 
12.. 33 
12.31 
12.29 
12.26 
12.21 
12.22 
12.20 
12.18 
12.16 


12.14 
12.12 
12.10 
12.07 
12.05 
12.03 
12.01 
11.99 
11.97 
11.95 


9.994620 
.991600 
.994-530 
.991.560 
.99454  ) 
.994519 
.994190 
.991479 
.994459 
.994433 

9.994413 
.994393 
.991377 
.994357 
.991336 
.991316 
.994295 
.994274 
.9942.54 
.994233 

9.994212 
.994191 
.994171 
.9941.50 
.994129 
.994103 
.994037 
.994060 
.994045 
.994024 

9.994003 
.993932 
.993960 
.993939 
.993913 
.993597 
.993375 
.903354 
.993332 
.993311 

9.993739 
.993763 
.993746 
.993725 
.993703 
.993631 
.993660 
.99.3633 
.99.3616 
.993594 

9.993572 
.993550 
.993528 
.993.506 
.993434 
.993462 
.993140 
.993413 
.99.3396 
.993374 
.993351 


.33 
.33 
.34 
.31 
.34 
.34 
.34 
.34 
.34 
.34 

.34 
.34 
.34 
.31 
.34 
.34 
.34 
.34 
.35 
.35 

.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

.35 
.35 
.•35 
..35 
.36 
.36 
.36 
.36 
.36 
.36 

..36 
..36 
.36 
.36 
..36 
.36 
.36 
.36 
.36 
.36 

.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 


9.199713 
.200529 
.201315 
.2021.59 
.202971 
.203732 
.204592 
.205400 
.206207 
.207013 

9.207317 
.203619 
.209420 
.210220 
.211013 
.211815 
.212611 
.213405 
.214193 
.214939 

9.215780 
.216563 
.217356 
.218142 
.213926 
.219710 
.220492 
.221272 
.222052 
.222830 

9.223607 
.224332 
.225153 
.225929 
.226700 
.227471 
.22^239 
.229007 
.229773 
.230539 

9.231302 
.232065 
.232326 
.233586 
.234345 
.235103 
.235359 
.236614 
.237.363 
.233120 

9.233372 
.239622 
.240371 
.211118 
.211365 
.242610 
.243354 
.244097 
.244839 
.245579 
.216319 


1.3.62 
13.. 59 
13.57 
13.54 
13.52 
13.49 
13.47 
13.45 
13.42 
13.40 

13.33 
13.35 
13.33 
13.31 
13.23 
13.26 
13.24 
13.21 
13.19 
13.17 

13.15 
13.12 
13.10 
1.3.03 
1.3.06 
13.03 
13.01 
12.99 
12.97 
12.95 

12.92 
12.90 
12.83 
12.36 
12.84 
12.82 
12.79 
12.77 
12.75 
12.73 

12.71 
12.69 
12.67 
12.65 
12.63 
12.60 
12.  .58 
12.56 
12.  .54 
12.52 

12.. 50 
12.43 
12.46 
12.44 
12.42 
12.40 
12.33 
12.36 
12.34 
12.32 


0.800237 
.799471 
.7936.55 
.797341 
.797029 
.796213 
.795403 
.794600 
.793793 
.792937 

0.792183 
.791331 
.790530 

.739780 
.788982 
.788185 
.787339 
.736595 
.785302 
.78501 1 

0.784220 
.783432 
.782644 
.781858 
.731074 
.7^0290 
.779503 
.778723 
.777943 
.777170 

0.776393 
.775618 
.774844 
.774071 
.773300 
.772529 
.771761 
.770993 
.770227 
.769461 

0.763693 
.767935 
.767174 
.766414 
.765655 
.764397 
.764141 
.763336 
.762632 
.761880 

0.761128 
.760378 
.759629 
.753332 
.758135 
.757390 
.756646 
.75.5903 
.755161 
.754421 
.753631 


60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 


40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 

4 
3 
2 
1 
0 


99^ 


184 

10^ 


TABLE    XIII. 


LOGARITHMIC    SINES, 


169! 


M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
lo 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
4t 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


9.239670 
.2403S6 
.241101 
.241814 
.242526 
.243237 
.243947 
.244656 
.245363 
.246069 

9. 246  r  75 
.247478 
.24S181 
.24S3S3 
.219583 
.250282 
.250980 
.251677 
.252373 
.253067 

9.253761 
.2.54453 
.255144 
.255834 
.256523 
.25721 1 
.257893 
.258583 
.259263 
.259951 

9.260633 
.261314 
.261994 
.262673 
.263351 
.264027 
.264703 
.26.5377 
.266r)51 
.266723 

9.267395 
.263065 
.263734 
.269402 
.270069 
.270735 
.271400 
.272064 
.272726 
.273338 

9.274049 
.274708 
.275367 
.276025 
.276631 
.2773.37 
.277991 
.278645 
.279297 
.279943 
.230599 


Cosino. 


D.  1". 


11.93 
11.91 
11.89 
11.87 
11.35 
11.83 
11.81 
11.79 
11.77 
11.75 

11.73 
11.71 
11.69 
11.67 
11.65 
11.63 
11.61 
11.59 
11.58 
11.56 

11.54 
11. .52 
11.50 
11.43 
11.46 
11.44 
11.42 
11.41 
11.39 
11.37 

11.35 
11.33 
11.31 
11.30 
11.23 
11.26 
11.24 
11.22 
11.20 
11.  L9 

11.17 
11.15 
11.13 
11.12 
11.10 
11.08 
11.06 
11.05 
11.03 
11.01 

10.99 
10.93 
10.96 
10.94 
10.92 
10.91 
10.89 
10.87 
10.86 
10.S4 

D.  1". 


Cosine. 


9.993351 
.993329 
.99.3307 
.99.3284 
.993262 
.993240 
.99.3217 
.993195 
.993172 
.993149 

9.993127 
.993104 
.993081 
.993059 
.993036 
.993013 
.992990 
.992967 
.992944 
.992921 

9.992898 
.992375 
.992852 
.992329 
,992306 
.992783 
.992759 
.992736 
.992713 
.992690 

9.992666 
.992643 
.992619 
.992596 
.992572 
.992549 
.992525 
.992501 
.992478 
.992454 

9.992430 
.992406 
.992382 
.992359 
.992335 
.992311 
.992237 
.992263 
.9922.39 
.992214 

9.992190 
.992166 
.992142 
.992118 
.992093 
.992069 
.992044 
.992020 
.991996 
.991971 
.991947 


D.  1". 


.37 
.37 
.37 
.37 
.37 
.37 
.38 
.38 
.38 
.33 

.38 
.38 
.38 
.33 
.38 
.38 
.33 
.38 
.38 
.33 

.38 
.38 
.39 
.39 
.39 
39 
39 
39 
39 
39 

39 

39 

39 

39 

.39 

.39 

.39 

.39 

.40 

.40 

.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 
.40 

.40 
.40 
.40 
.41 
.41 
.41 
.41 
.41 
.41 
.41 


Sine.    D.  1".  Cotang 


Tang. 


9.246319 
.247057 
.247794 
.243530 
.249264 
.249993 
.250730 
.251461 
.252191 
.252920 

9.253648 
.254374 
.255100 
.255824 
.256547 
.257269 
.257990 
,258710 
.259429 
,260146 

9.260863 
.261578 
.262292 
.263005 
.263717 
.264428 
.265133 
.265347 
.266555 
.267261 

9.267967 
,263671 
.269375 
.270077 
.270779 
,271479 
,272178 
.272876 
.273573 
.274269 

9.274964 
,275653 
,276351 
.277043 
.277734 
.278424 
.279113 
.279301 
.230438 
.281174 

9.281858 
.232542 
.233225 
.283907 
.234.588 
.235263 
.235947 
.236624 
.287301 
.237977 
.2386.52 


D,  1". 


12.30 
12.28 
12.26 
12.24 
12.22 
12.20 
12.18 
12.17 
12.15 
12.13 

12.11 
12.09 
12.07 
12.05 
12.03 
12.01 
12.00 
11.98 
11.96 
11.94 

11.92 
11.90 
11.89 
11.87 
11.85 
11.83 
11.81 
11.79 
11.78 
11.76 

11.74 
11,72 
11,70 
11.69 
11.67 
11,65 
11.64 
11.62 
11.60 
11.58 

11.57 
11.55 
11.53 
11.51 
11.50 
11.48 
11.46 
11.45 
11.43 
11,41 

11,40 
11,38 
11.36 
11,35 
11.33 
11.31 
11.30 
11.23 
11.26 
11.25 


D,  1". 


Cotang, 


0.753681 
,752943 
.752206 
.751470 
,750736 
,750002 
.749270 
,748539 
,747309 
,747080 

0.746352 
,745626 
,744900 
,744176 
,743453 
,742731 
,742010 
,741290 
,740571 
,7.39854 

0,739137 
,738422 
,737708 
,736995 
,736283 
.735572 
,734862 
,7341.53 
,73.3445 
,732739 

0,732033 
,731329 
.7.30625 
.729923 
,729221 
,728521 
,727822 
,727124 
,726427 
,725731 

0.725036 
,724342 
,723649 
,722957 
,722266 
,721576 
,720887 
.720199 
,719512 
,718826 

0,718142 
.717453 
,716775 
716093 
,715412 
,714732 
.714053 
,713376 
,712699 
.712023 
,711348 


Tang, 


1003 


»$»* 


COSINES,    TANGENTS,    AND    COTANGENTS. 


M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
l.j 
16 
17 
IS 
19 

20 
21 
22 
23 
21 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 


54 
55 
56 
57 
53 
59 
60 

M. 


185 

168° 


Sine. 


9.230599 
.231243 
.231897 
.232.544 
.233190 
.233336 
.284480 
.235124 
.235766 
.236103 

9.237043 

.237633 
.238326 
.283964 
.239600 
.290236 
.290370 
.291504 
.292137 
.292763 

9.293399 
.291029 
.294658 
.295286 
.29.5913 
.296.539 
.297161 
.297733 
.293412 
.299031 

9.299555 
.300276 
.300395 
.391514 
..3021.32 
.302743 
.303364 
..303979 
.304593 
.305207 

9.305319 
.3061.30 
.307041 
.307650 
.3082,59 
.3)8367 
.309474 
.310030 
.310635 
.311239 

9.311893 
.312495 
.31.3097 
.31.3693 
.314297 
.314397 
.315495 
.316092 
.316039 
.317234 
.317879 


D.  1". 


10.82 
10.81 
10.79 
10.77 
10.76 
10.74 
10.72 
10.71 
10.69 
10.67 

10.66 
10.64 
10.63 
10.61 
10.59 
10.53 
10.56 
10.  .55 
10.53 
10.51 

10.50 
10.43 
10.47 
10.45 
10.43 
10.42 
10.40 
10.39 
10.. 37 
10.. 36 

10.. 34 
10.33 
l(l.31 
10.  .30 
10.23 
10.20 
10.25 
10.23 
10.22 
10.20 

10.19 
10.17 
10.16 
10.14 
10.13 
10.12 
10.10 
10.09 
10.07 
10.06 

10.04 

10.03 

10.01 

10.00 

9.93 

9.97 

9.96 

9.94 

9.93 

9.91 


Co-sine.   D.  1" 


Cosine. 


9.991947 
.991922 
.991897 
.991873 
.99134^ 
.991823 
.991799 
.991774 
.991749 
.991724 

9.991699 
.991674 
.991619 
.991624 
.991.599 
.991574 
.991549 
.991524 
.991498 
.991473 

9.99144^ 
.991422 
.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 

9.991193 
.991167 
.991141 
.991115 
.991090 
.991064 
.991038 
.991012 
.9909S6 
.990960 

9.990934 
.990908 
.990332 
.990355 
.990829 
.990303 
.990777 
.990750 
.990724 
.990697 

9.990671 
.990645 
.990618 
.990.591 
.990565 
.990538 
.990511 
.990485 
.9904.58 
.990431 
.990404 


Sine. 


D.  1". 


.41 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
42 

.42 
.42 
.42 

.42 
.42 
.42 
.42 
.42 
.42 
.42 

.42 
.42 
.42 
.42 
.42 
.43 
.43 
.43 
.43 
.43 

.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 
.43 

.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 

.44 
.44 
.44 
.44 
.44 
.44 
.45 
.45 


D.  1". 


Tang. 


9.23^652 
.239326 
.239999 
.290671 
.291312 
.292013 
.292632 
.293350 
.29liM7 
.294634 

9.295349 
.296  )13 
.296677 
.297339 
.298001 
.298662 
.2993i!2 
.299930 
.300033 
.301295 

9.301951 
.302607 
.303261 
.303914 
..304.-)67 
.305218 
.305369 
.306519 
.307168 
.307816 

9.303463 
.309109 
.3097.54 
.310399 
.311042 
.311635 
.312327 
.312968 
.31.3603 
.314247 

9.314335 
.315523 
.316159 
.316795 
.3174.30 
.318064 
.313697 
.319.330 
.319961 
.320592 

9.321222 
..321S51 
.322479 
.323106 
.3237.33 
.324358 
.324933 
.32.5607 
.32H231 
.3268.53 
.327475 


Cotang. 


D.  1". 


11.23 
11.22 
11.20 
11.18 
11.17 
11.15 
11.14 
11.12 
11.11 
11.09 

11.07 
11. '^6 
11.04 
11.03 
11.01 
11.00 
10.98 
10.97 
10.95 
10.93 

10.92 
10.90 
10.89 
10.87 
10.^6 
10.84 
10.^3 
10.81 
10.8') 
10.78 

10.77 
10.76 
10.74 
10.73 
10.71 
10.70 
10.68 
10.67 
10.65 
10.64 

10.62 
10.61 
10.60 
10.58 
10.. 57 
10.55 
10.54 
10.  .53 
10.51 
10.50 

10.48 
10.47 
10.46 
10.44 
10.43 
10.41 
10.40 
10.39 
10.37 
10.36 


D.  1". 


Cotang. 

0.711343 
.710674 
.710001 
.709329 
.708658 
.707987 
.707318 
.706650 
.705933 
.705316 

0.704651 
.703987 
.703323 
.702661 
.701999 
.701333 
.700678 
.700020 
.699362 
.698705 

0.693049 
.697.393 
.696739 
.6960*6 
.6954.33 
.691782 
.694131 
.693431 
.692832 
.692184 

0.691537 

.690591 
.690246 
.639601 
.638953 
.633315 
.637673 
.63703-2 
.636392 
.685753 

0.635115 
.684477 
.633841 
.633205 
.632570 
.631936 
.631303 
.630670 
.630039 
.679403 

0.673778 
.673149 
.677521 
.676>!94 
.676267 
.675042 
.675017 
.674.393 
.673769 
.673147 
.672525 


Tang. 


lOlo 


r8« 


186 


TABLE    Xlll         L05AR1TJMIC    SINES, 


16TC 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
S 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
5S 
59 
60 

M. 


Si 


ine 


9.317879 
.318473 
.319066 
.3196.0 
.320249 
.320840 
.321430 
.322019 
.322607 
.323194 

9.323780 
..324366 
.324950 
.32.5534 
.326117 
.326700 
.327281 
.327862 
.328442 
.329021 

9.329.399 
..330176 
.330753 
..331329 
.331903 
.332478 
.333051 
.333624 
.334195 
.334767 

9.355337 
.335906 
.336475 
.337043 
.337610 
.333176 
.338742 
.339-307 
.339871 
.340431 

9.340996 
.341558 
.342119 
.342679 
.ai32.39 
.343797 
.3443.55 
.344912 
.34.5469 
.346024 

9.346579 
.347134 
.347637 
.348240 
.348792 
.349343 
.349393 
.350443 
.350992 
.351540 
.352088 

Cosine. 


D.  1". 


Cosine. 


9.90 
9.83 
9.87 
9.86 
9.34 
9.  S3 
9.81 
9.-0 
9.79 
9.77 

9.76 
9.75 
9.73 
9.72 
9.70 
9.69 
9.63 
9.66 
9.65 
9.64 

9.62 
9.61 
9.60 
9.53 
9.57 
9.. 56 
9.54 
9.53 
9.52 
9.50 

9.49 
9.43 
9.46 
9.45 
9.44 
9.43 
9.41 
9.40 
9.39 
9.37 

9.36 
9.35 
9.34 
9.-32 
9.31 
9.30 
9.29 
9.27 
9.26 
9.25 

9.24 
9.22 
9.21 
9.20 
9.19 
9.17 
9.16 
9.15 
9.14 
9.13 

D.  1". 


9.990404 
.990378 
.990351 
.990324 
.990297 
.990270 
.y9ri243 
.990215 
.990183 
.990161 

9.990134 
.990107 
.990079 
.9900.52 
.990025 
.989997 
.989970 
.939942 
.939915 
.989837 

9.989860 
.989832 
.989804 
.939777 
.939749 
.989721 
.989693 
.939665 
.939637 
.989610 

9.989-532 
.989553 
.939525 
.989497 
.939469 
.989441 
.939413 
.939385 
.9393.56 
.989328 

9.989300 
.989271 
.989243 
.939214 
.939186 
.9391.57 
.939123 
.989100 
.989071 
.939042 

9.989014 
.988985 
.988956 
.938927 
.933393 
.933369 
.938340 
.933311 
.988782 
.988753 
.'933724 

Sine. 


D.  1". 


.45 
.45 
.45- 
.45 
.45 
.45 
.45 
.45 
.45 
.45 

.45 
.45 
.46 
.46 
.46 
.46 
.46 
.46 
.46 
.46 

.46 
.46 
.46 
.46 
.46 
.46 
.46 
.47 
.47 
.47 

.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 

.47 
.47 
.47 

.48 
.48 
.48 
.48 
.48 
.43 
.48 

.48 

.48 
.48 
.48 
.48 
.48 
.48 
.43 
.49 
.49 

D.  1". 


Tang.  I  D.  1". 


9.327475 
.328095 
.323715 
.329334 
.3299.53 
.3.30.570 
.331137 
.331803 
.332413 
.333033 

9.333646 
.3.34259 
.334871 
.335482 
.336093 
.336702 
.337311 
.337919 
.338527 
.339133 

9.3397.39 
.340344 
.340943 
.341552 
.342155 
.3427.57 
343358 
3439.38 
.3445.58 
.345157 

9.345755 
..346.353 
..346949 
.347.545 
.343141 
.348735 
.349329 
..349922 
.3.50514 
.351 IG6 

9.351697 
..352287 
.352376 
.3.53465 
.3540.53 
..3:34640 
.355227 
.355813 
..356398 
.356982 

9.357566 
.3.58149 
.358731 
.3.59313 
.3.59893 
.360474 
.361053 
.3616.32 
.362210 
.362787 
■  363.364 

Cotang. 


10.35 
10.33 
10.32 
10.31 
10.29 
10.23 
10.27 
10.25 
10.24 
10.23 

10.21 
10.20 
10.19 
10.17 
10.16 
10.15 
10.14 
10.12 
10.11 
10.10 

10.03 
10.07 
10.06 
10.05 
10.03 
10.02 
10.01 
10.00 
9.98 
9.97 

9.96 
9.95 
9.93 
9.92 
9.91 
9.90 
9.88 
9.87 
9.66 
9.85 

9.84 
9.82 
9.81 
9.80 
9.79 
9.78 
9.76 
9.75 
9.74 
9.73 

9.72 
9.70 
9.69 
9.63 
9.67 
9.66 
9.65 
9.63 
9.62 
9.61 

D.  1". 


Cotang. 


0.672525 
.671905 
.671285 
.670666 
.670047 
.669430 
.663813 
.663197 
.667582 
.666967 

0.666354 
.665741 
.665129 
.664518 
.663907 
.663293 
.662689 
.662031 
.661473 
.660867 

0.660261 
.659656 
.6590.52 
.6.5S448 
.657845 
.657243 
.656642 
.6.56042 
.655442 
.654843 

0.654245 
.6.53647 
.6.5.3051 
.652455 
.651859 
.651265 
.6.50671 
.650078 
.649486 
.648594 

0.648303 
.647713 
.&47124 
.646.535 
,645947 
.645360 
.644773 
.644187 
.643602 
.643018 

0.6424:34 
.641851 
.641269 
.640687 
.640107 
.639.526 
.638947 
.638368 
.637790 
.637213 
.636636 

Tang. 


103? 


77= 


COSINES     TANGENTS,    AND    COTANGENTS. 


13^ 

M. 

0 
1 
2 
3 
4 
5 
6 
7 


181 
1663 


Sine. 


D.  1". 


10 
U 
12 
13 

14 
15 
16 
17 

IS 
19 

20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 
31 
32 
33 
31 
3o 
3d 
37 
3S 
39 

40 
41 
42 
43 
41 
45 
46 
47 
4S 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


9.352'H8 
.352635 
.353181 
.353726 
.354271 
.3.')  IS  1 5 
.355353 
.355901 
.3.56113 
.356934 

9.357524 
.35SU64 
.358603 
.359141 
.359673 
.360215 
.360752 
.361287 
.361822 
.362356 

9.362389 
.353122 
.363954 
..364485 
.365016 
.365546 
.366075 
.3^)66)4 
.367131 
.367659 

9.  .368 1 85 
.368711 
.369236 
369761 
.370285 
.370SOS 
.371330 
.371852 
.372373 
.372894 

9..37;M14 
.373933 
.374452 
.374970 
.375487 
.376003 


,377035 
.377549 
.3780S3 

9.373577 
.379089 
.379601 
.380113 
.380621 
.381134 
.381643 
.382152 
.3S2661 
.3S3163 
.383675 


Cosine. 


9.11 
9.10 
9.09 
9.08 
9.07 
9.05 
9.04 
9.03 
9.02 
9.01 

8.99 
8.98 
8.97 

8.96 
8.95 
8.91 
8.92 
8.91 
8.90 
8.89 

8.83 
8.87 
8.86 
8.84 
8.33 
8.82 
8.81 
8.80 
8.79 
8.78 

8.76 
8.75 
8.74 
8.73 
8.72 
8.71 
8.70 
8.69 
8.63 
8.66 

8.65 
8.61 
8.03 
8.62 
8.61 
8.60 
•  8.59 
8.. 53 
8.57 
8.56 

8.55 
8.53 
8.52 
8.51 
8.. 50 
8.49 
8.48 
8.47 
8.46 
8.45 


D.l". 


9.938724 
.988695 
.938666 
.938636 
.938607 
.933573 
.988.548 
.938519 
.938489 
.938460 

9.988430 
.9^3401 
.988371 
.933342 
.933312 
.913282 
.9>3252 
.938223 
.938193 
.938163 

9.933133 
.933103 
.988073 
.988043 
.938013 
.937933 
.937953 
.937922 
.937892 
.987862 

9.9873.32 
.937801 
.987771 
.937740 
.937710 
.937679 
.937649 
.937618 
.937.588 
.987557 

9.937526 
.937496 
.937465 
.937434 
.957403 
.937372 
.937341 
.987310 
.937279 
.987248 

9.937217 
.937186 
.987155 
.937124 
.937092 
.987061 
.937030 
.936998 
.936967 
.936936 
.936904 


M. 


103  2 


Cosine. 


D.  1". 


Tang. 


Sine. 


.49 
.49 
.49 
.49 
.49 
.49 
.49 
.49 
.49 
.49 

.49 
.49 
.49 
.50 
.50 
..50 
.50 
.50 
.50 
.50 

.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.51 

.51 
.51 
.51 
.51 
.51 
.51 
.51 
.51 
.51 
.51 

.51 
.51 
.51 
.51 
.51 
.52 
.52 
.52 
.52 
.52 

.52 
..52 
.52 
.52 
..52 
.52 
.52 
52 
.52 
.52 


D.  1" 


9.363364 
.363910 
.361515 
.365f)90 
.365664 
.366237 
.366810 
.367332 
.367953 
.363524 

9.369094 
.369663 
•370232 
.370799 
.371367 
.371933 
.372499 
.373064 
.373629 
.374193 

9.374756 
.375319 
.375881 
.376442 
.377003 
.377563 
.378122 
.373631 
.379239 
.379797 

9.3803.54 
.330910 
.331466 
.332020 
.382575 
.333129 
.333632 
.384231 
.334786 
.385337 

9.335388 
.386433 
.386937 
.387536 
.333031 
.383631 
.339178 
.339724 
.390270 
.390315 

9.391360 
.391903 
.392447 
.392939 
.39.3531 
.394073 
.394614 
.395154 
.395694 
.396233 
.396771 


Cotang. 


9.60 
9.59 
9.58 
9.57 
9.. 55 
9.54 
9.53 
9.52 
9.51 
9.50 

9.49 
9.48 
9.47 
9.45 
9.44 
9.43 
9.42 
9.41 
9.40 
9.39 

9.33 
9.37 
9.36 
9.-35 
9.33 
9.32 
9.31 
9.30 
9.29 
9.23 

9.27 
9.26 
9.25 
9.24 
9.23 
9.22 
9.21 
9.20 
9.19 
9.18 

9.17 
9.16 
9.15 
9.14 
9.12 
9.11 
9.10 
9.09 
9.08 
9.07 

9.06 

9.(je 

9.04 
9.03 
9.02 
9.01 
9.00 
8.99 
8.93 
8.97 


0.636636 
.636060 
.635435 
.634910 
.0.34336 
.633763 
.633190 
.632618 
.632047 
.631476 

0.630906 
.630337 
.629768 
.629201 
.628633 
.628067 
.627501 
.0269.36 
.626371 
.625807 

0.625244 
.624681 
.624119 
.623558 
.622997 
.6221.37 
.621873 
.621319 
.620761 
.620203 

0.619640 
.619090 
.618.534 
.617930 
.617425 
.616371 
.616318 
.615766 
.615214 
.614663 

0.614112 
.61.3562 
.613013 
.612464 
.611916 
.611369 
.610822 
.610276 
.609730 
.609185 

0.6)8640 
.608097 
.607553 
.607011 
.6()6469 
.605927 
.605386 
.604846 
.604306 
.603767 
.6')3229 


M. 

60 
59 

53 


I 


D.  1".  I  Cotang.   D.  1" 


54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 

38 

37 

36 

35  ! 

34 

33 

32 

31 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


Tang. 


M. 


7Bi 


1«8 
140 


TABLE    XIII.       LOGARITHMIC    MNES, 


165C^ 


M. 

~0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

]0 
II 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


9.333675 
.334182 
.334637 
.335192 
.335697 
.336201 
.3S6704 
.337207 
.337709 
.333210 

9.  .33  37 11 
.339211 
.33971 1 
.390210 
.390703 
.391206 
.391703 
.392199 
.392695 
.393191 

9.393635 
..394179 
.394673 
.395166 
.395653 
..3961.50 
.395641 
.397132 
.397621 
.398111 

9.398600 
.399038 
.399575 
.400062 
.400549 
.401035 
.401520 
.402005 
.402439 
.402972 

9.403455 
.403938 
.404420 
.404901 
.405382 
.405362 
.406341 
.406320 
.407299 
.407777 

9.403254 
.408731 
.409207 
.409682 
.410157 
.410632 
.411106 
.411579 
.412052 
.412524 
.412996 

Cosine. 


D.  1". 


8.44 
8.43 
8.42 
8.41 
8.40 
8.-39 
8.38 
3.37 
3.36 
8.35 

8.34 
8.-33 
8.32 
8.31 
8.30 
8.29 
8.28 
8.27 
8.26 
8.25 

8.24 
8.2:3 
8.22 
8.21 
8.20 
8.19 
8.18 
8.17 
8.16 
8.15 

8.14 
8.13 
8.12 
8.11 
8.10 
8.09 
8.08 
8.07 
8.06 
8.05 

8.04 
8.03 
8.02 
8.01 
8.00 
7.99 
7.93 
7.97 
7.96 
7.96 

7.95 
r.94 
7.93 
7.92 
7.91 
7.90 
7.89 
7.83 
7.87 
7.86 


Cosine. 


9.9869C4 
.936873 
.936841 
.986809 
.986778 
.986746 
.986714 
.986633 
.986651 
.936619 

9.9S6587 
.986555 
.986523 
.936491 
.936459 
.936427 
.936.395 
.986-363 
.986-331 
.986299 

9.9=6266 
.986234 
.986202 
.986169 
.9361.37 
.986104 
.936072 
.936039 
.986007 
.93.5974 

9.98.5942 
,985909 
.935876 
.935843 
.985811 
.985773 
.985745 
.985712 
.985679 
.935646 

9.985613 
.985580 
.98.5547 
.98-5514 
.98-5480 
.935447 
.935414 
.985381 
.98-5347 
.985314 

9.935230 
.935247 
.93-5213 
.985130 
.985146 
.935113 
.935079 
.935045 
.93501 1 
.934978 
.934944 

Sine. 


D.  1". 


.53 
.53 
.53 
.53 
.53 
.53 
.53 
.53 
.53 
.53 

.53 
.53 
.53 
.53 
.53 
..54 
.54 
.54 
.54 
.54 

.54 
.54 
M 
.54 
.54 
..54 
.54 
.54 
.54 
.54 

.54 
.55 
..55 
..55 
.55 
.55 
.55 
.55 


.-55 
.55 
.55 
.55 
.55 
.55 
..56 
.56 
.56 
.56 

.56 
.56 
.56 
.56 
.5f 
M 
.56 
..56 
.56 
.56 

D.  1". 


Tang. 


9.396771 
.397309 
.397846 
.393333 
.398919 
.399455 
399990 
.400524 
.401058 
.401591 

9.402124 
.402656 
.403187 
.403718 
.404249 
.404778 
.405308 
.405836 
.406364 
.406392 

9.407419 
.407945 
.408471 
.408996 
.409521 
.410045 
.410.569 
.411092 
.411615 
.412137 

9.4126.53 
.413179 
.413699 
.414219 
.414738 
.415257 
.415775 
.416293 
.416810 
.417-326 

9.417842 
.4183-58 
.418873 
.419337 
.419901 
.420415 
.420927 
421440 
4219-52 
.422463 

9.422974 
.423434 
.423993 
.424503 
.425011 
.425519 
.426027 
.426-534 
.427041 
.427.547 
.428052 

Cotang. 


D.  1". 


8.96 
8.96 
8.95 
8.94 
8.93 
8.92 
8.91 
8.90 
8.89 
8.88 

8.87 
8. 86 
8.85 
8.84 
8.83 
8.82 
8.81 
8.80 
8.79 
8.78 

8.77 
8.76 
8.75 
8.75 
8.74 
8.73 
8.72 
8.71 
8.70 
8.69 

8.63 
8.67 
8.66 
8.65 
8.65 
8.64 
8.63 
8.62 
8.61 
8.60 

8.59 

8.58 
8.57 
8.56 
8.56 
8.-55 
8.-54 
8.53 
8.52 
8.51 

8.50 
8.49 
8.49 
8.48 
8.47 
8.46 
8.45 
8.44 
8.43 
8.43 

D.  1". 


Cotang 

M. 

60 

0.603229 

.602691 

59 

.602154 

58 

.601617 

57 

.601081 

56 

.600545 

55 

.600010 

54 

.599476 

53 

.593942 

52 

.598409 

51 

0.597376 

50 

.597344 

49 

.596313 

48 

.596232 

47 

.595751 

46 

.595222 

45 

.594692 

44 

.5941&4 

43 

.5936.36 

42 

.593103 

41 

0.592531 

40 

.5920.55 

39 

.591529 

38 

.591004 

37 

.590479 

36 

.589955 

35 

.539431 

34 

.588908 

33 

.583385 

32 

.587863 

31 

0.537342 

30 

.586821 

29 

.586301 

28 

.535781 

27 

.585262 

26 

.534743 

25 

.584225 

24 

.583707 

23 

..583190 

22 

.582674 

21 

0.582158 

20 

.581642 

19 

.581127 

18 

.-580613 

17 

.580099 

16 

.579535 

15 

..579073 

14 

.578560 

13 

.578043 

12 

.577537 

11 

0.577026 

10 

.576516 

9 

.576007 

8 

.57.5497 

7 

.574989 

6 

.574481 

5 

.573973 

4 

.573466 

3 

.572959 

2 

.572453 

1 

.571948 

0 
M. 

Tang. 

1040 


T«i 


COSINES,    TANGENTS,    AND    COTANGENTS. 


189 


M. 

0 
1 

2 
3 
4 
5 
6 
7 


Sine. 


9.412996 
.413467 
.413933 
.414408 
.414S78 
.415347 
.415815 
.416283 
.416751 
.417217 


D.l" 


10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 

32 

33 

.34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

4S 

49 

50 
51 

52 
53 

54 
55 
56 

57 
58 

ro 

60 


9.417634 
.418150 
.418615 
.419079 
.419544 
.420007 
.420470 
.420933 
.421395 
.421857 

9.422318 
.422773 
.4232.33 
.423697 
.424156 
.424615 
.425073 
.425530 
.425987 
.426443 

9.426899 
.427354 
.427809 
.428263 
.428717 
.429170 
.429623 
.430075 
.430.527 
.430978 

9.431429 
.431879 
.4.32329 
.432778 
.433226 
.433675 
.434122 
.434569 
.435016 
.435462 

9.43.5903 
.436353 
.436793 
.437242 
.437636 
.438129 
.43^572 
.439014 
.439456 
.439397 
.440338 


Cosine. 


7.85 
7.84 
7.84 
7.83 
7.82 
7.S1 
7.80 
7.79 
7.78 
7.77 

7.76 
7.75 
7.75 
7.74 
7.73 
7.72 
7.71 
7.70 
7.69 
7.68 

7.67 
7.67 
7.66 
7.65 
7.6-4 
7.63 
7.62 
7.61 
7.61 
7.60 

7.59 
7.58 
7.57 
7.56 
7.55 
7.55 
7.53 
7.52 
7.52 
7.51 

7.50 
7.49 
7.49 
7.48 
7.47 
7.46 
7.45 
7.44 
7.44 
7.43 

7.42 

7.41 

7.40 

7.40 

7.39 

7.38 

7.37 

7.36 

7  36 

7.35 


D.  1". 


9.984944 
.984910 
.984876 
.934842 
.984308 
.984774 
.934740 
.934706 
.934672 
.934638 

9.984603 
.984569 
.984535 
.984500 
.984466 
.984432 
.934397 
.934363 
.984328 
.934294 

9.934259 
.934224 
.934190 
.934155 
.984120 
.984085 
.984050 
.984015 
.933931 
.983946 

9.9S3911 
.983875 
.933840 
.933805 
.983770 
.933735 
.983700 
.983664 
.983629 
.983594 

9.983553 
.933523 
.983487 
.9834.52 
.983416 
.983381 
.983345 
.983309 
.98.3273 
.983238 


M.       Cosine.    I    D.  1". 


9.98.3202 
.983166 
.983130 
.983094 
.983058 
.933022 
.982936 
.982950 
.932914 
.932378 
.982342 


Tang. 


.56 

.57 
.57 

.57 
.57 
.57 
.57 
.57 
.57 
.57 

.57 
.57 
.57 
.57 

.57 
.57 

.58 
.58 
.58 
.58 

.58 
.58 
.58 
.58 
.58 
.58 
.53 
.53 
.53 
.58 

.58 
.58 
.59 
.59 
.59 
.59 
.59 
.59 
.59 
.59 

59 
.59 
.59 
.59 
.59 
.59 
..59 
.60 
.60 
.60 

.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 


9.428052 
.428558 
.429062 
.429566 
.430070 
.430573 
.431075 
.431577 
.432079 
.432580 

9.433080 
.433580 
.434080 
.434579 
.435078 
.435576 
.436073 
.436570 
.437067 


Sine. 


D.  1". 


.437563 

9.438059 
.4335.54 
.439048 
.439543 
.440036 
.440529 
.441022 
.441514 
.442006 
.442497 

9.442988 
.443479 
.443968 
.444458 
.444947 
.445435 
.445923 
.446411 
.446898 
.447384 

9.447870 
.443356 
.443841 
.449326 
.449810 
.450294 
.4.50777 
.451260 
.451743 
.452225 

9.452706 
.453187 
,453668 
.4.54148 
.4.54628 
.455107 
.455586 
.456064 
.456542 
.457019 
.457496 


D.r 


Cotang. 


8.42 

8.41 

8.40 

8.39 

8.38 

8.38 

8.37 

8.36 

8.35 

8.34 

8.33 
8.33 

8.32 
8.31 
8.30 
8.29 

8.28 
8.28 
8.27 
8.26 

8.25 

8.24 
8.24 
8.23 
8.22 
8.21 
8.20 
8.20 
8.19 
8.18 

8.17 
8.16 
8.16 
8.15 
8.14 
8.13 
8.13 
8.12 
8.11 
8.10 

8.09 
8.09 

8.08 
8.07 
8.06 
8.06 
8.05 
8.04 
8.03 
8.03 

8.02 

8.01 

8.00 

8.00 

7.99 

7.98 

7.97 

7.97 

7.96 

7.95 


M. 


0.571948 
.571442 
.570933 
.570434 
.569930 
.569427 
.568925 
.563423 
.567921 
.567420 

0.566920 
.566420 
.565920 
.565421 
.564922 
.564424 
.563927 
.5^3430 
.562933 
,562437 

0.561941 
.561446 

.560952 
.560457 
.559964 
.5.59471 

.556978 
.558486 
.557994 
.557.503 

0.5.57012 
.556521 
.556032 
.555542 
.555053 
.554565 
.554077 
.553589 
.553102 
.552616 

0.552130 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 


.fc)0 


Cotang. 


D.  1". 


1644 

.551159 
.550674 
.550190 
.549706 
.549223 
.543740 
.543257 
.547775 

0.547294 
.546313 
.546332 
.545852 
.545372 
.544893 
.544414 
.543936 
.543458 
.542981 
.542.504 


Tang. 


50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
_0^ 

M. 


105° 


7*0 


190 

160 


TABLE    Xlll.       LOGARITHMIC    SINES, 


163f 


M. 

0 
1 
2 
3 

4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
IS 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
3! 
32 
33 
31 
35 
3G 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


9.440333 
.440778 
.441213 
.4416.53 
.442096 
,442535 
.442973 
.41.3410 
.443347 
.444231 

9.444720 
.445155 
.445590 
.446025 
.4464-59 
.446393 
.447326 
.447759 
.443191 
.443623 

9.4490.54 
.449435 
.449915 
450345 
.450775 
.451204 
.451632 
.4.52060 
.452483 
.4.52915 

9.4.53342 
.453763 
.451194 
.454619 
.45.5044 
.45.5469 
.455393 
.4.56316 
.456739 
.457162 

9.457534 
.453006 
.453427 
.453S4S 
.459263 
.4.59633 
.460103 
.460527 
.460946 
.461364 

9.461782 
.462199 
.462616 
.463932 
.463448 
.463864 
.454279 
.464694 
.465103 
.465522 
.465935 

Cosine. 


D  1". 


7.34 
7.33 
7.32 
7.31 
7.31 
7.30 
7.29 
7.23 
7.27 
7.27 

7.26 
7.25 
7.24 
7.24 
7.23 
7.22 
7.21 
7.20 
7.20 
7.19 

7.18 
7.17 
7.17 
7.16 
7.15 
.14 
.13 
.13 
.12 
.11 


7. 
7. 

7. 
7. 
7. 


7.10 
7.10 
7.09 
7.08 
7.07 
7.07 
7.06 
7.05 
7.04 
7.04 

7.03 
7.02 
7.01 
7.01 
7.00 
6.99 
6.98 
6.98 
6.97 
6.96 

6.96 
6.95 
6.94 
6.93 
6.93 
6.92 
6.91 
6.90 
6.90 
6.S9 


Cosine. 


9.982842 
.982305 
.982769 
.9327.33 
.982696 
.932660 
.9326^4 
.9»25'^7 
.982551 
.982514 

9.9S2477 
.932441 
.982404 
.932367 
.932331 
.982294 
.9322.57 
.932220 
.982183 
.982146 

9.9>2109 
.932072 
.9820.35 
.931998 
.981961 
.931924 
.9818^6 
.981849 
.931812 
.981774 

9.931737 
.931700 
.931662 
.931625 
.981587 
.931549 
.931512 
.981474 
.931436 
.981399 

9.981361 
.981323 
.931235 
.931247 
.931209 
.931171 
.9311.33 
.931095 
.931057 
.981019 

9.980931 
.930942 
.930904 
.930366 
.930827 
.930789 
.930750 
.930712 
.930673 
.930635 
.980.596 


D.  1".  I  Sine. 


D.  1". 


.60 
.60 
.61 
.61 
.61 
.61 
.61 
.61 
.61 
.61 

.61 
.61 
.61 
.61 
.61 
.61 
.61 
.62 
.62 
.62 

.62 
.62 
.62 
62 
.62 
.62 
.62 
.62 
.62 
.62 

.62 
.62 
.63 
.63 
.63 
.63 
.63 
.63 
.63 
.63 

.63 
.63 
.63 
.63 

.63 
.63 
.63 
.64 
.64 
.64 

.64 
.64 
.64 
.64 
.64 
.64 
.64 
.64 
.64 
.64 


Tang. 


9.4574,16 
.457973 
.453449 
.453925 
.459400 
.459875 
.460.349 
.460323 
.461297 
.461770 

9.462242 
.462715 
.463b:6 
.463658 
.464128 
.464.599 
.46.5069 
.465539 
.466008 
.466477 

9.4R6945 
.467413 
.4678^0 
.468347 
.463314 
.469280 
.469746 
.470211 
.470676 
.471141 

9.471605 
.472069 
.472.532 
.472995 
.473457 
.473919 
.474381 
.474342 
.475303 
.475763 

9.476223 
.476633 
.477142 
.477601 
.478059 
.473517 
.478975 
.479432 
.479839 
.430345 

9.480301 
.431257 
.431712 
.432167 
.482621 
.483075 
.433.529 
.433932 
.434435 
.434837 
.435339 


D.  1".   Cotang. 


D.  1". 


7.94 
7.91 
7.93 
7.92 
7.91 
7.91 
7.90 
7.89 
7.83 
7.83 

7.87 
7.>6 
7.86 
7.85 
7.84 
7.  S3 
7.83 
7. 82 
7.^1 
7.81 

7.30 
7.79 
7.78 
7.73 
7.77 
7.76 
7.76 
7.7o 
7.74 
7.74 

7.73 
7.72 
7.71 
7.71 
7.70 
7.69 
7.69 
7.63 
7.67 
7.67 

7.66 
7.65 
7.65 
7.64 
7.63 
7.63 
7.62 
7.61 
7.61 
7.60 

7.59 
7.  .59 
7.53 
7.57 
7.57 
7.  .56 
7.55 
7.55 
7.54 
7.53 


D.  1". 


Cotang. 


0.542504 
.542027 
.541.551 
.541075 
.540600 
.540125 
..539651 
.539177 
.538703 
..533230 

0.537758 

.5372^5 

.5.36314 

.5.36;M2 

.535372 

.535401  ' 

.534931 

.15.34461 

.533992 

.533523 

0.533055 
.532587 
.5.32120 
.5316.53 
.531186 
.530720 
.530254 
.529739 
..529324 
.528859 

0.523395 
.527931 
.527463 
.527005 
.526-543 
.526031 
.52.5619 
.525153 
.524697 
.524237 

0.523777 
.523317 
..522353 
.522399 
..521941 
..521433 
.521025 
.520.563 
.520111 
.519655 

0.519199 
.518743 
.518283 
.517833 
.517379 
.516925 
.516471 
.516018 
.515565 
.515113 
.514661 


Tang. 


^060 


73^ 


COSINES, 


TANGENTS,  AND  COTANGENTS. 


191 


M. 


a 
6 

7 
8 
9 

in 
11 

12 
13 
14 
1-3 
16 
17 
18 
19 

20 
21 
22 
23 

24 
25 
26 
27 
2S 
23 

30 

31 

32 

33 

34 

35 

36 

37 

SS 

39 

40 

41 

42 

43 

44 

4', 

46 

47 

4S 

49 

50 
51 


Sine. 

9.465935 
.466348 
.466761 
.467173 
.4675S5 
.467996 
.463407 
.463317 
.469227 
.469637 

9.470046 
.470455 
.470363 
.471271 
.471679 
.472036 
.472492 
.472393 
.473301 
.473710 

9.474115 
.474519 
.474923 
.475327 
.475730 
.476133 
.476.536 
.476933 
.477310 
.477741 

9.478142 
.478542 
.478942 
.479342 
.479741 
.430140 
.430539 
.480937 
.431334 
.431731 

•9.432128 
.432525 
.432921 
.483316 
.483712 
.484107 
.434501 
.434395 
.435239 
.485632 

9.436075 
.436467 


D  1".  I  Cosine.   D-  1".   Tang. 


52 

.436S60 

53 

.437251 

54 

.437643 

55 

.483034 

56 

.433424 

57 

.488314 

53 

.439204 

59 

.489593 

60 

.439932 

M. 

Cosine. 

6  83 
6.88 
6.87 
6.86 
6.85 
6.85 
6.84 
6.83 
6.83 
6.82 

6.81 

6.81 

6.80 

6.79 

6.78 

6.78 

6.77 

6.76 

6.76 

6.75 

6.74 
6.74 
6.73 
6.72 
6.72 
6.71 
6.70 
6.69 
6.69 
6.63 

6.67 
6.67 
6.66 
6.65 
6.65 
6.64 
6.63 
6.63 
6.62 
6.61 

6.61 
6.60 
6.59 
6.59 


6.57 
6.57 
6.56 
6.55 
6.55 

6.-54 
6.. 54 
6.53 
6.52 
6.52 
6.51 
6.50 
6.50 
6.49 
6.48 


9.950596 
.930553 
.930519 
.930430 
.980412 
.980403 
.930364 
.930325 
.930236 
.930247 

9.980208 
.930169 
.980130 
,930091 
.980052 
.980012 
.979973 
.979934 
.979395 
.979855 

9.979316 
.979776 
.979737 
.979697 
.979653 
.979613 
.979579 
.979539 
.979499 
.979459 

9.979420 
.979330 
.979310 
.979300 
.979260 
.979220 
.979130 
.979140 
.979100 
.979059 

9.979019 
.978979 
.978939 
.978898 
.978353 
.978317 
.978777 
.973737 
.978696 
.978055 

9.978615 
.978574 
.978533 
.978493 
.978452 
.973411 
.978370 
.978329 
.978233 
.978217 
.973206 


D.  1" 


D.  1".  I  Sine 


9.485339 

.485791 
.436242 
.4'^6693 
.487143 
.437593 
.483043 
.488492 
.483941 
.489390 

9.439833 
.490236 
.490733 
.491180 
.491627 
1  .492073 
.492519 
.492965 
.493110 
.493354 

9.494299 
.494743 
.495136 
.495630 
.496073 
.496515 
.496957 
497399 
.497841 
.493232 

9.493722 
.499163 
.499603 
.500042 
.500431 
.500920 
.5013.59 
.501797 
..502235 
.502672 

9.503109 
50.3546 
503932 
.504418 
.504354 
.505239 
.50.5724 
.506159 
..506593 
.507027 

9.507460 
.507893 
.503326 
.503759 
.509191 
.509622 
.5100.54 
.510435 
.510916 
.511346 
.511776 


Cotang. 


D.  1".  I  Cotang. 


7.53 
7.52 
7.51 

7.51 
7.50 
7.  .50 
7.49 

7.43 
7.43 
7.47 

7.46 
7.46 
7.45 
7.44 
7.44 
7.43 
7.43 
7.42 
7.41 
7.41 

7.40 

7.39 

7.39  . 

7.33 

7.33 

7.37 

7.36 

7.36 

7.35 

7.34 

7.-34 
7.33 
7.33 
7.32 
7.31 
7.31 
7.30 
7.30 
7.29 
7.23 

7.23 
7.27 
7.27 
7.26 
7.25 
7.25 
7.24 
7.24 
7.23 
7.23 

7.22 
7.21 
7.21 
7.20 
7.20 
7.19 
7.18 
7.18 
7.17 
7.17 


0.514661 
.514209 
.513758 
.513307 
.512357 
.512407 
.5119.57 
.511503 
.5110.59 
.510610 

0.510162 
.509714 
.509267 
.503320 
.505373 
.507927 
.507431 
.507035 
.506590 
.506146 

0.505701 
.505257 
.504314 
.504370 
.50-3927 
.503435 
.50-3043 
.502601 
.5021-59 
.501718 

0.501278 
.500^37 
.500397 
.499958 
.499519 
.499030 
.493641 
.493203 
.497765 
.497328 

0.496391 
.496454 
.496018 
.495532 
.495146 
.494711 
.494276 
.493341 
.493407 
.492973 


M. 

60 

59 
53 
57 
56 


D.  1". 


Ci492540 
.492107 
.491674 
.491211 
.490309 
.490373 
.4899 16 
.439515 
.489034 
.483654 
,438224 


54 
53 


50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 

39 

33 

37  I 

36 

35 

34 

33 

32 

31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 

8 
7 
0 


Tang. 


lor^ 


4 
3 
2 
1 
0 

M. 


7a« 


192 

183 


TABLE    XIII.       LOGARITHMIC    SINES, 


161<; 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


9.4S9932 
.490371 
.490759 
.491147 
.491535 
.491922 
.492303 
.492695 
.493031 
.493466 

9.493351 
.494236 
.494621 
.495005 
.49.!;333 
.495772 
.496154 
.496.537 
.496919 
.497301 

9.497632 
.493064 
.493444 
.493325 
.499214 
.499534 
.499963 
.500342 
.500721 
.501099 

9.. 50 14  76 
.5015.54 

..50-^231 
..502607 
.502934 
.503360 
.5037.3.J 
.504110 
.504435 
.504560 

9.  .505234 
.505603 
..505931 
..506354 
..506727 
.507099 
.507471 
.507343 
..503214 
.503535 

9.503956 
.509326 
.509696 
.510065 
.510434 
.510303 
.511172 
.511540 
.511907 
.512275 
.512642 

C:«ine. 


D.  1". 


6.43 
6A7 
6.46 
6.46 
6.45 
6.45 
6.44 
6.43 
6.43 
6.42 

6.41 
6.41 
6.40 
6.39 
6.. 39 
6.  .33 
6.33 
6..37 
6.36 
6.36 

6.35 
6.34 
6.3^4 
6.33 
6.33 
6.32 
6.31 
6.31 
6.30 
6.30 

6.29 
6.23 
6.23 
6.27 
6.27 
6.26 
6.25 
6.25 
6.24 
6.24 

6.23 
6.22 
6.22 
6.21 
6.21 
6.20 
6.19 
6.19 
6.13 
6.13 

6.17 
6.16 
6.16 
6.15 
6.15 
6.14 
6.14 
6.13 
6.12 
6.12 

D.  1". 


Cosine. 


9.973206 
.973165 
.973124 
.973033 
.973042 
.973001 
.977959 
.977913 
.977377 
.977835 

9.977794 
.977752 
.977711 
.977669 
.977623 
.977536 
.977^544 
.977503 
.977461 
.977419 

9.977377 
.977335 
.977293 
.977251 
.977209 
.977167 
.977125 
.977033 
.977041 
976999 

9.976957 
.976914 
.976372 
.976330 
.976737 
.976745 
.976702 
.976660 
.976617 
.976574 

9.976532 
.976439 
.976446 
.976404 
.976:361 
.976313 
.976275 
.976232 
.976139 
.976146 

9.976103 
.976060 
.976017 
.975974 
.975930 
.975357 
.975344 
.975300 
.975757 
.975714 
■975670 

Sine. 


D.  1". 


.63 
.69 
.69 
.69 
.69 
.69 
.69 
.69 
.69 
.69 

.69 
.69 
.69 
.69 
.69 
.69 
.70 
.70 
.70 
.70 

.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 

.70 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 
.71 

.71 
.71 

.71 
.71 
.71 
.72 
.72 
.72 
.72 
.72 

.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 

D.  1". 


Tang. 


9.511776 
.512206 
.512635 
.513064 
513493 
.513921 
.514349 
.514777 
.515204 
.515631 

9.516057 
.516434 
.516910 
.517.335 
.517761 
.513156 
.515810 
.519034 
.5194.53 
.519382 

9.520305 
.520723 
.521 151 
.521573 
..521995 
.522417 
.522333 
.523259 
.52.3630 
.524100 

9.524-520 
.524940 
.525359 
.525778 
.526197 
..526615 
.527033 
.527451 
.527863 
.528285 

9.525702 
.-529119 
..529.535 
..529951 
.5.30-366 
..5-30781 
..531196 
..531611 
.53202-5 
.5-32439 

9.-532353 
.533266 
.5.33679 
.534092 
.534-504 
.53^916 
.53.5323 
.535739 
.-5361.50 
.-5-36561 
.5-36972 

Cotang. 


D.  1". 


7.16 
7.16 
7.15 
7.14 
7.14 
7.13 
7.13 
7.12 
7.12 
7.11 

7.10 
7.10 
7.09 
7.09 
7.03 
7.03 
7.07 
7.07 
7.06 
7.05 

7.05 
7.04 
7.04 
7.03 
7.03 
7.02 
7.02 
7.01 
7.01 
7.00 

6.99 
6.99 
6.93 
6.98 
6.97 
6.97 
6.96 
6.96 
6.G5 
6.95 

6.94 
6.94 
6.93 
6.93 
6.92 
6.91 
6.91 
6.90 
6.90 
6.89 

6.39 
6.33 
6.83 
6.87 
6.87 
6.86 
6.86 
6.85 
6. 85 
6.34 

D.  1". 


Cotang. 


0.438224 
.437794 
.437365 
.456936 
.486507 
.456079 
.435651 
.435223 
.434796 
.434369 

0.48-3943 
.433516 
.45-3090 
.452665 
.452239 
.431514 
.481.390 
.430966 
.430.542 
.430113 

0.479695 
.479272 

.478349 
.473427 
.478005 
.477553 
.477162 
.476741 
.476320 
.475900 

0.475450 
.475060 
.474641 
.474222 
.47-3303 
.473355 
.472967 
.472;!^  9 
.472132 
.471715 

0.47129S 
.47033! 
.470465 
.470049 
.4696:34 
.469219 
.465504 
.463389 
.467975 
.467561 

0.467147 
.466734 
.466321 
.465908 
.465496 
.465034 
.464672 
.464261 
.46-33^50 
.46:34:39 
.463023 

Tang 


M. 


1085 


7J' 


COSINES,    TANGENTS,    AND    C0TANC4ENTS. 


193 

160C 


M. 

0 
1 

2 
3 

4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 
lo 
16 
17 
IS 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 

42 

43 
44 

45 
46 
47 
48 
49 

50 
51 
52 
53 
51 
55 
56 
57 
5S 
59 
60 

M. 


Sine. 


D.  1". 


9.512642 
.513009 
.513375 
.513741 
.514107 
.514472 
.514837 
.515202 
.515566 
.515930 

9.516294 
.516657 
.517020 


.517745 
.518107 
.518463 
.513829 
.519190 
.519551 

9.519911 
.520271 
.520631 
.520990 
.521349 
.521707 
.522066 
.522424 
.522781 
.52:3133 

9.523495 
.523352 
.524203 
.524564 
.524920 
.525275 
.525630 
.525934 
..526339 
.526693 

9.527046 
.527400 
.527753 
.523105 
.523453 
.523310 
.529161 
.529513 
.529364 
.530215 

9.530r565 
.530915 
.531265 
.531614 
.531963 
.532312 
.532661 
.533009 
.533357 
.533701 
.531052 

Cosine. 


Cosine. 


6.11 
6,11 
6.10 
6.09 
6.09 
6.03 
6.03 
6.07 
6.07 
6.06 

6.05 
6.05 
6.04 
6.01 
6.03 
6.03 
6.02 
6.02 
6.01 
6.00 

6.00 
5.99 
5.99 
5.93 
5.98 
5.97 
5.97 
5.96 
5.95 
5.95 

5.94 
5.94 
5.93 
5.93 
5.92 
5.92 
5.91 
5.90 
5.90 
5.89 

5.89 

5.88 
5.88 
5.87 
5.87 
.5.86 
5.86 
5.85 
5.85 
5.84 


5.33 

5.82 
5.82 
5.81 
5.81 
5.30 
5.30 
5.79 
5.79 


D.  1". 


9.975670 
.975627 
.975533 
.975539 
.975496 
.975452 
.975403 
.975365 
.975321 
.975277 

9.975233 
.975189 
.975145 
.975101 
.975057 
,975013 
.974969 
.974925 
.974880 
.974336 

9.974792 
.974748 
.974703 
.974659 
.974614 
.974570 
.974525 
.974481 
.974436 
.974391 

9.974347 
.974302 
.974257 
.974212 
.974167 
.974122 
.974077 
.974032 
.973937 
.973942 

9.973397 
.973852 
.973307 
.973761 
,973716 
.973671 
.973625 
.973530 
.973535 
.973489 

9.973444 
.973393 
.973352 
.973307 
.973261 
.973215 
.973169 
.973124 
.973078 
.973032 
.972936 


.73 
.73 


Tang. 


D.  1". 


Sine. 


.73 
.73 
.73 
.73 
.73 
.73 

.73 
.73 
.73 
.73 
.73 
.74 
.74 
.74 
.74 
.74 

.74 

.74 

.74 
.74 
.74 
.74 
.74 
.74 
.74 
.75 

.75 
.75 

.75 
.75 
.75 

.75 
.75 
.75 
.75 
.75 

.75 
.75 
.75 
.75 
.76 
.76 
.76 
.76 
.76 
.76 

.76 
.76 
.76 
.76 

.76 
•.76 
.76 
.76 

.77 
.77 


9.536972 
.537382 
.537792 
,533202 
.53861 1 
.539020 
.539429 
.539837 
.540245 


D.  1" 


.0 


40653 

9.541061 
,541463 
.541875 
.542231 
.542638 
.543094 
.543499 
.543905 
.544310 
,544715 

9.545119 

,545524 
.545928 
,546331 
.546735 
,547138 
.547540 
.547943 
.548345 
,548747 

9.. 549 149 
.549550 
..549951 
,550352 
.550752 
.551153 
.551552 
.551952 
.5523:51 
.552750 

9.553149 
.553548 
.553946 
.554344 
.554741 
.555139 
.555536 
.555933 
.556329 
.556725 

9.5.57121 
.557^17 
.557913 
..558303 
.558703 
..559097 
.559491 
.559335 
.560279 
.560673 
..561066 


6.84 
6.33 
6.83 
6.82 
6.82 
6.81 
6.81 
6.80 
6.80 
6.79 

6.79 
6.78 
6.78 
6.77 
6.77 
6.76 
6.76 
6.75 
6.75 
6.74 

6.74 
6.73 
6.73 
6.72 
6.72 
6.71 
6.71 
6.70 
6.70 
6.69 

6.69 
6.68 
6.63 
6.67 
6.67 
6.67 
6.66 
6.66 
6.65 
6.65 

6.64 
6.64 
6.63 
6.63 
6.62 
6.62 
6.61 
6.61 
6.60 
6.60 

6.59 
6.59 
6.59 
6.58 
6.58 
6.57 
6.57 
6.56 
6.56 


D.  1".  Cotang. 


Cotang. 

0.463023 
.462618 
.462203 
.461798 
.461339 
.460980 
.460571 
.460163 
,459755 
.459347 

0.458939 
.458532 
,458125 
.457719 
.457312 
.4.56906 
.456501 
.4.56095 
.455690 
.455285 

0.454881 
.454476 
.4.54072 
.4536P9 
.453265 
.452362 
.452460 
,452057 
.451655 
,451253 

0.450351 
.450450 
.450049 
.449643 
.449248 
.448847 
.443443 
,443018 
.447649 
.447250 

0.446351 
,446452 
,446054 
,445656 
.445259 
,444861 
.444464 
.441067 
.443671 
.443275 

0.442379 
.442483 
.442037 
.441692 
.441297 
,440903 
,440509 
.440115 
.439721 
.439327 
.433934 


M. 


60 
59 
53 
57 
56 
55 
34 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


D.  1". 


Tang. 


M. 


1090 


7©: 


194 


TABLE    XIII. 


LOGARITHMIC    SINES, 


159<J 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
IS 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
4S 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

M. 


Sine. 


9.534052 
..5-34399 
.534745 
.53.5092 
.53.5438 
.535783 
.536129 
.536474 
.536318 
.537163 

9.537507 
.537851 
.533194 
.538538 
.533880 
.539223 
..5.39.565 
.539907 
.540249 
.540590 

9.540931 
.541272 
.541613 
..541953 
.542293 
.5426.32 
.542971 
.543310 
.543649 
.543987 

9.544325 
.544663 
.545000 
545338 
.545674 
.546011 
.546347 
..546633 
.547019 
.547354 

9.547639 

.548024 
.543359 
.543693 
..549027 
.549360 
.549693 
.5.50026 
.5.50359 
.550692 

9.  .55 1024 
.551356 
.551687 
..552018 
.552349 
.552630 
.5.53010 
.55.3341 
.553670 
.554000 
.554329 

Cosine. 


D.  1". 


5.78 
5.73 
5.77 
5.77 
5-.  76 
5.76 
5.75 
5.75 
5.74 
5.74 

5.73 
5.73 
5.72 
5.71 
5.71 
5.70 
5.70 
5.69 
5.69 
5.68 

5.63 
5.67 
5.67 
5.66 
5.66 
5.65 
5.65 
5.64 
5.64 
5.63 

5.63 
5.62 
5.62 
5.61 
5.61 
5.60 
5.60 
5.59 
5.59 
5.58 

5.58 
5.57 
5.57 
5.56 
5.56 
5.55 
5.55 
5.55 
5.54 
5.54 

5.53 
5.53 
5.52 
5.52 
5.51 
5.51 
5.50 
5.50 
5.49 
5.49 

D.  1' . 


Cosine. 


9.972986 
.972940 
.972394 
.972348 
.972302 
.972755 
.972709 
.972663 
.972617 
.972570 

9.972524 
.972478 
.972431 
.972335 
.972333 
.972291 
.972245 
.972193 
.972151 
.972105 

9.972053 
.972011 
.971964 
.971917 
.971870 
.971323 
.971776 
.971729 
.971632 
.971635 

9.971588 
.971.540 
.971493 
.971446 
.971398 
.971351 
.971303 
.9712.56 
,971203 
.971161 

9.971113 
.971066 
.971018 
.970970 
.970922 
.970874 
.970827 
.970779 
.970731 
.970633 

9.970635 
,970586 
.970538 
.970490 
.970442 
.970394 
.970345 
.970297 
.970249 
.970200 
.970152 

Sine. 


D.  1". 


.77 
.77 
.77 
.77 
■  .77 
.77 
.77 
.77 
.77 
.77 

.77 
.77 
.73 
.78 
.78 
.78 
.78 
.78 
.78 
.78 

.78 
.78 
.78 
.73 
.78 
.78 
.78 
.79 
.79 
.79 

.79 
.79 
.79 
.79 
.79 
.79 
.79 
.79 
.79 
.79 

.79 
.80 
.80 
.80 
.80 
.80 
.80 
.80 
.80 
.80 

.80 
.80 
.80 
.80 
.80 
.81 
.81 
.81 
.81 
.81 

D.  1". 


Tang. 


9.561066 
.561459 
.561851 
.562244 
.562636 
.563023 
.563419 
.563311 
.564202 
.564593 

9.564933 
.565373 
.565763 
.566153 
.566542 
.566932 
.567320 
.567709 
.563093 
.563486 

9.563373 
..569261 
.569643 
.570035 
.570422 
.570309 
.571195 
.571531 
.571967 
.572352 

9.572733 
.573123 
.573507 
.573392 
,574276 
.574660 
.575044 
.575427 
.575310 
.576193 

9.576576 
.576959 
.5773-11 
.577723 
.578104 
.578486 
.578367 
.579243 
.579629 
.530009 

9.580339 
.580769 
.581149 

.581523 
.531907 
.532236 
.532665 
.533044 
.533422 
.533300 
.584177 

Cotang. 


D.  1". 


6.55 
6.54 
6.. 54 
6.54 
6.53 
6.53 
6.52 
6.52 
6.51 
6.51 

6.. 50 
6.50 
6..50 
6.49 
6.49 
6.48 
6.48 
6.47 
6.47 
6.46 

6.46 
6.46 
6.45 
6.45 
6.44 
6.44 
6.43 
6.43 
6.43 
6.42 

6.42 
6.41 
6.41 
6.40 
6.40 
6.40 
6.39 
6.39 
6.38 
6.33 

6.37 
6.37 
6.37 
6.36 
6.36 
6.35 
6.35 
6.34 
6.34 
6.34 

6.33 
6.33 
6.32 
6.32 
6.32 
6.31 
6.31 
6.30 
6.30 
6.30 

D.  1". 


Cotang. 


0.438934 

.4.33541 
.433149 
,437756 
.437364 
.436972 
.436.-5S1 
.436139 
.435793 
.435407 

0.43.5017 
.434627 
.434237 
.4.33847 
.4.33453 
.4.33068 
.4.32680 
.432291 
.431902 
.431514 

0.431127 
.430739 
.430352 
.429965 
.429578 
.429191 
.428805 
.428419 
.423033 
.427648 

0.427262 

.426377 
.426493 
.426103 
.425724 
.425340 
.424956 
.424573 
.424190 
.423307 

0.423424 
.423041 
.422659 
.422277 
.421896 
.421514 

-.421133 
.420752 
.420371 
.419991 

0.419611 
.419231 
.418351 
.418472 
.413093 
.417714 
.417335 
.416956 
.416578 
.416200 
.415823 

Tang. 


IIOO 


603 


COSINES,    TANGENTS,    AND    COTANGENTS. 


1580 


M. 


Sine. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
4S 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


9.554329 
.554653 
.55411^7 
.555315 
.555643 

.  .555971 
.556299 
.r 56626 
.556953 
.557280 

9.557606 
.557932 

..558253 
.55858  ! 
.5.58909 
.5592:34 
.559558 
.559883 
.560207 
.560531 

9.560855 
..561178 
.561501 
.561824 
..562146 


D.  1''. 


.0 


62468 
.562790 
.563112 
.563433 
.563755 

9.564075 
.564396 
.564716 
.565036 
.565356 
..565676 
.565995 
..566314 
..566632 
..566951 

9.567269 
.567587 
.567904 
.568222 
.568539 
.568856 
.56917^ 
.569438 
..569804 
.570120 

9.570435 
.570751 
.571066 
..571.3S0 
.571695 
.572009 
,572323 
572636 
572950 
.573263 
..573575 


Cosine. 


D.  1". 


5.48 
5.48 
5.47 
5.47 
5.46 
5.46 
5.45 
5.45 
5.44 
5.44 

5.44 

5.43 
5.43 
5.42 
5.42 
5.41 
5.41 
5.40 
5.40 
5.39 

5.39 
5.38 
5.33 
5.37 
5.37 
5.37 
5.36 
5.36 
5.35 
5.-35 


5.34 
5.33 
5.33 
5.32 
5.32 
5.32 
5.31 
5.31 
5.30 

5.. 30 
5.29 
5.29 
5.28 
5.28 
5.28 
5.27 
5.27 
5.26 


5.25 
5.24 
5.24 
5.24 
5.23 
5.23 
5.22 
5.22 
5.21 


9.970152 
.970103 
.970055 
.970006 
.969957 
.969909 
.969'-60 
.969811 
.969762 
.969714 

9.969665 
.969616 
.969567 
.969518 
.969469 
.969420 
,969370 
.969321 
.969272 
,969223 

9.969173 
,969124 
.969075 
.969025 
.968976 
.968926 
.968877 
.968827 
.968777 
.968728 

9.96S678 
.968628 
.968578 
.968523 
.968479 
.968429 
.968379 
.963329 
.963278 
.963228 

9.968178 
.968128 
.963078 
.963027 
.967977 
.967927 
.967876 
.967826 
.967775 
.967725 

9.967674 
.967624 
.967573 
.967.522 
.967471 
.967421 
.967370 
.967319 
.967268 
.967217 
.967166 


M.  I  Cosine.  I  D.  1". 


Tang. 


D.  1". 


Sine. 


.81 
.81 
,81 
.81 
81 
.81 
.81 
.81 
.81 
,81 

,82 
.82 
.82 
.82 
.82 
.82 
.82 
.82 
.82 
.82 

.82 
.82 
.82 
.82 
.83 
,83 
.83 
,83 
,83 
,83 

.83 

,83 
,83 
.83 
.83 
.83 
.83 
.83 
.84 
.84 

84 
.84 
.84 
.84 
.84 
.84 
.81 
.84 
.84 
.84 

.84 
.84 
.85 
.85 
.85 
.85 
.85 
.85 
.85 


9.584177 
.584555 
.584932 
..585309 
.585686 
.586062 
.536439 
.5%815 
.587190 
.587566 

9.587941 
.588316 
.588691 
..589066 
.589440 
.589814 
.590188 
.590562 
.590935 
.591308 

9.591681 
.592054 
,592426 
.592799 
.593171 
.593.542 
.593914 
.594285 
,594656 
,595027 

9.595393 
,595768 
.596138 

..596508 
.596878 
..597247 
I  .597616 
I  .597985 
.598354 
.598722 

9.599091 
.599459 
.599827 
.600194 
.600.562 
.600929 
.601296 
.601663 
.602029 
.602395 

9.602761 
.603127 
.603493 
.603858 
.604223 
.604583 
.601953 
,605317 
.605682 
.606046 
.606410 


Cotaiig.   M. 


D.  1".   Cotang. 


6.29 
6.29 
6.28 
0.28 
6.28 
6.27 
6.27 
6.26 
6.26 
6.26 

6.25 
6.25 
6.24 
6.24 
6.24 
6.23 
6.23 
6.22 
6.22 
6.22 

6.21 
6.21 
6.20 
6,20 
6.20 
6.19 
6.19 
6.18 
6.18 
6.18 

6.17 
6.17 
6.16 
6.16 
6.16 
6.15 
6.15 
6.15 
6.14 
6.14 

6.13 
6.13 
6.13 
6.12 
6.12 
6.12 
6.11 
6,11 
6.10 
6.10 

6.10 

6.09 
6.09 
6.09 
6.08 
6.03 
6.07 
6.07 
6.07 
6.06 


0. 41.5^23 
.415445 
.41.5068 
.414691 
.414314 
.413938 
.413561 
.413185 
.412810 
,412434 

0.412059 
.411684 
,411309 
,410934 
.410560 
.410186 
.409812 
.409438 
.409065 
.408692 

0.408319 
.407946 
.407574 
.407201 
.406829 
.406458 
.406086 
.405715 
.405344 
.404973 

0.404602 
.404232 
.403^62 
.4034'..2 
.403122 
.402753 
.402384 
.402015 
.401646 
.401278 

0.400909 
.400541 
,400173 
.399806 
.399433 
.399071 
.398704 
.398337 
.397971 
.397605 

0.397239 
.396873 
.396507 
.396142 
.395777 
.395412 
.395047 
,394683 
.394318 
.393954 
.393590 


1).  1". 


Tang.   M 


60 
59 

68 


56 


53 

52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


111 


68^ 


*-W9rr^^9^m^\^^ww-r  #^f  .A^V^^tV/jT^^AV 


196 

933 


TABLE    XIII.       LOGARITHMIC    SI^'ES, 


157' 


M. 

0 
1 
2 
3 

4 
5 
6 

7 
8 
9 

10 
II 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 
31 
32 
33 
34 
3-5 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
4S 
49 

50 
51 
52 
53 
54 
55 
58 
57 
53 
59 
60 


Sine. 


9.573575 
.573333 
.574200 
.574512 
.574S24 
.575136 
.57.5147 
.575753 
.576069 
.576379 

9.576639 
.576999 
.577309 
.577613 
.577927 
.573236 
.573545 
.573853 
.579162 
.579470 

9.579777 
.530035 
.530392 
.530699 
.531005 
.531312 
.531613 
.531924 
.532229 
.532535 

9.532340 
.533145 
.533449 
.533754 
.581053 
.534361 
..531665 
.534963 
..535272 
.535574 

9. 535S77 
..536179 

.  .536132 
.536733 
,537035 
.537336 
.537633 
.537939 
.583239 
.533590 

9.533390 
.539190 
.539439 
.539789 
.590033 
.590337 
.593636 
.500934 
.591232 
.591530 
.591373 


D.  1".   Cosine. 


M.   Cosine.   D.  1". 


5.21 
5.2( 
5.2C 
5.20 
5,19 
5.19 
5.13 
5.13 
5.17 
5.17 

5.17 
5.16 
5.16 
5. 15 
5.15 
5.14 
5.14 
5.14 
5.13 
5.13 

5.12 
5.12 
5.11 
5.11 
5.11 
5.10 
5.10 
5.09 
5.09 
5.09 

5.03 
5.03 
5.07 
5.07 
5.06 
5.06 
5.06 
5.05 
5.05 
5.04 

5.04 
5.04 
5.03 
5.03 
5.02 
5.02 
5.01 
5.01 
5.01 
5.00 

5.00 
4.99 
4.99 
4.99 
4.93 
4.93 
4.97 
4.97 
4.97 
4.96 


9.967166 
.967115 
.967064 
.967013 
.966961 
.966910 
.966359 
.966303 
.966756 
.966705 

9.966653 
.966602 
.966550 
.966499 
.966447 
.966395 
.966iH 
.966292 
.966240 
.966133 

9.966136 
.966035 
.966933 
.965981 
.965929 
.965376 
.965324 
.965772 
.965720 
.965663 

9.96.5615 
.965563 
.965511 
.965453 
.96^5406 
.965a53 

-.965301 
.965243 
.965195 
.965143 

9.965090 
.965037 
.961934 
.964931 
.961379 
.964326 
.961773 
.961720 
.961666 
.961613 

9.961560 
.964507 
.964454 
.964400 
.964:347 
.961294 
.961240 
.961187 
.964133 
.961030 
.961026 


D.  1". 


Sine. 


.85 
.85 
.85 
.85 
.85 
.35 
.86 
.86 
.86 
.36 

.86 
.86 
.86 
.86 
.86 
.86 
.56 
.86 
.86 
.86 

.87 

.87 
.87 
.37 
.87 
.87 
.87 
.87 
.87 
.87 

.87 
.37 
.87 
.87 
.83 
.83 
.83 
.88 
.83 
.83 

.83 
.83 
.83 
.83 
.83 
.33 
.33 
.83 
.89 
.39 

.89 
.89 
.89 
.89 
.39 
.39 
.89 
.89 
.89 
.89 


Tang. 


D.  1". 


9.605110 
.606773 
.607137 
.607.500 
.6' 17363 
.603225 
.6034533 
.603950 
.609312 
.609674 

9.610036 
.610397 
.610759 
.611120 
.611430 
.611341 
.6122)1 
.612561 
.612921 
.613231 

9.613641 
.614000 
.6143.59 
.614713 
.615077 
.6154.35 
.615793 
.616151 
.616509 
.616367 

9.617224 
.617.532 
.617939 
.613295 
.613652 
.619033 
.619364 
.619720 
.620076 
.623432 

9.620737 
.621142 
.621497 
.621352 
.622207 
.622561 
.622915 
.623269 
.623623 
.623976 

9.624330 
.624633 
.6250.36 
.62-5333 
.625741 
.626093 
.626445 
.626797 
.627149 
.627-501 
.627352 


Cotang. 


D.  1". 

6.06 
6.06 
6.05 
6.05 
6.05 
6.04 
6.04 
6.03 
6.03 
6.03 

6.02 
6.02 
6.02 
6.01 
6.01 
6.01 
6.00 
6.00 
6.00 
5.99 

5.99 
5.93 
5.93 
5.93 
5.97 
5.97 
5.97 
5.96 
5.96 
5.96 

5.95 
5.95 
5.95 
5.94 
5.94 
5.94 
5.93 
5.93 
5.93 
5.92 

5.92 
5.92 
5.91 
5.91 
5.91 
5.90 
5.90 
5.90 
5.89 
5.89 

5.89 
5.83 
5.83 
5.83 
5.87 
5.87 
5.87 
5.86 
5.86 
5.86 


D.  1". 


Cotang. 

0.393590 
.393227 
.392363 
.392500 
.392137 
.391775 
.391412 
..3910-50 
.390633 
.390326 

0.339964 
.339603 
.339241 
.333330 
.333520 
.338159 
.337799 
.337439 
.387079 
.336719 

0.336359 
..336000 
.33.5641 
.33-5232 
.334923 
.384565 
.384207 
.33:3349 
.333491 
.333133 

0.332776 
.332413 
.332061 
.331705 
.331:343 
.330992 
.330636 
.330230 
.379924 
.379563 

0.379213 
.378353 
.373503 
.373143 
.377793 
.377439 
.377035 
.376731 
.376:377 
.376024 

0.375670 
.375317 
.374964 
.374612 
.3742.59 
.373907 
.373555 
.373203 
.372351 
.372499 
.372148 


Tang. 


iia^j 


67" 


COSINES,    TANGENTS,    AND    COTANGENTS. 


830 


\91 

15G3 


i. 

Sine. 

0 

9.591878 

1 

.59217G 

2 

..592-173 

3 

.592770 

4 

.5'.):{I67 

5 

..^.93363 

6 

.n936.j9 

7 

.593955 

8 

.594251 

9 

.591547 

D.  1". 


10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
2G 
27 
28 
29 

30 
31 
32 
33 
31 
35 
36 
37 
33 
39  j 

40 

4r 

42 
43 
44 
45 
46 
47 
4S 
49 

50 

51 

52 

53 

54 

55 

56 

57 

53 

59 

60 


ri3o 


9.594842 
.595137 
..59.5432 
..595727 
.596021 
.596315 
.5966;  )9 
.59(5903 
.597196 
.597490 

9.. 597783 
.593075 
.598363 
.593660 
.598952 
.599244 
.599.536 
.599827 
.600118 
.600409 

9.600700 
.601)990 
.6012-0 
.601570 
.601860 
.692 150 
.602439 
.602728 
.603017 
.603305 

9.603594 
.603382 
.6:14170 
.604457 
.604715 
.605032 
.605319 
.605606 
.605892 
.606179 


9.60r.'65 
.606751 
.607036 
.607322 
.607607 
.607892 
.608177 
.60-461 
.603745 
.609029 
.609313 

Cosine. 


4.96 
4.95 
4.95 
4.95 
4.94 
4.94 
4.93 
4.93 
4.93 
4.92 

4.92 

4.91 

4.91 

4.91 

4.90 

4.90 

4.89 

4.89 

4.S9 

4.88 

4.83 
4.83 
4.87 
4.S7 
4.86 
4.86 
4.86 
4.85 
4.85 
4.S4 

4.84 
4.84 
4.83 
4.83 
4.83 
4. 82 
4.82 
4.81 
4.81 
4.81 

4.80 
4.80 
4.79 
4.79 
4.79 
4.73 
4.78 
4.7.S 
4.77 
4.77 

4.76 
4.76 
4.76 
4.75 
4.75 
4.74 
4.74 
4.74 
4  73 
4.73 


D.  1". 


Cosine. 

9.964026 
.963972 
.963919 
.963365 
.963811 
.9(;37.j7 
.963704 
.9636.30 
.963596 
.963:542 

9.9634-13 
.963434 
.963379 
.963325 
.96327! 

'.■^3217 
.9631ftJ 
.9631(13 
.96;',(l.54 
.962999 

9.962945 
.962390 
.962336 
.962781 
.932727 
.962672 
.9G2;i7 
.962.562 
.962.503 
-.962453 

9.962398 
.962343 
.962288 
.962233 
.962178 
.962123 
.962067 
.962012 
.9619.57 
.961902 

9.961846 
.961791 
.9617.35 
.961630 
.961624 
.961569 
.961513 
.9614.53 
.961402 
.961346 

9.961290 
.961235 
.961179 
.96112! 
.961007 
.961011 
.960955 
.96:)V.)9 
.960343 
.96II7S6 
.96)730 

Sine. 


D.  1" 


Tang. 


.89 
.89 
.90 
.90 
.90 
.90 
.90 
.90 
.90 
.90 

.90 
.90 
.90 
.90 
.90 
.911 
.91 
.91 
.91 
.91 

.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.91 
.92 

.92 
.92 

.92 
.92 
.92 
.92 
.92 
.92 
.'M 
.92 

.92 
.92 
.92 
.93 
.93 
.93 
.93 
.93 
.93 
.93 

.93 
.93 
.93 
.93 
.93 
.93 
.93 
.94 
.94 
.94 


1).  1" 


9.627852 
.628203 
,<^28554 
.623905 
.629255 
.629606 
.629956 
.630306 
.630656 
.631005 

9.631355 
.631704 
.6.32053 
.6321(12 
.632750 
.633099 
.633447 
.633795 
.634143 
.634490 

9.634333 
.635185 
.635532 
.635379 
.636226 
.636572 
.636919 
.637265 
.637611 
.637956 

9.638302 
.633617 
.633992 
.639337 
.639632 
.640027 
.610371 
.640716 
.641060 
.6414(04 

9.641747 
.642091 
.642434 
.642777 
.643120 
.643463 
.643306 
.644148 
.644490 
.644332 

9.645174 
.64-5516 
.645S57 
.646199 
.616540 
.646361 
.647222 
.647562 
.617903 
.643243 
.643583 


D.  1", 


5.85 
5.85 
5.85 
5. 84 
5.54 
5. 84 
5.83 
5.83 
5.83 
5.82 

5.82 
5.82 
5.81 
5.81 
5.81 
5.80 
5.80 
5.80 
5.79 
5.79 

5.79 

5.78 
5.78 
5.78 


5.77 
5.77 

5.77 


Cotang. 


.76 
6 


o./ 


5.75 
5.75 
5.75 
5.74 
5.74 
5.74 
5.73 
5.73 
73 


Cotang. 


y, 

5.73 
5.72 
5.72 
5.72 
5.71 
5.71 
5.71 
5.70 
5.70 
5.70 

5.69 
5.69 
5.69 
5.69 
5.68 
5.68 
5.68 
5.67 
5.67 
5.67 


-|| 


0.372148 
.371797 
.371446 
.371095 
.370745 
.370394 
.370044 
.369694 
.369344 
.363995 

0.36S645 
.363296 
.367947 
.367593 
.367250 
.366901 
.366553 
.366205 
.365857 
.365510 

0.365162 
.364>15 
.364468 
.364121 
.363774 
.363423 
.363081 
.3627.35 
.362339 
.362044 

0.361698 
.361353 
.361008 
.360663 
.360318 
.3.59973 
.359629 
.359234 
.358940 
.358596 

0.358253 
.357909 


D.  1". 


.357223 

.356380 
.356537 
,356194 
,355852 
.355510 
.355163 

0.354826 
.3.544^4 
.3.54143 
.3533!  II 
.353460 
.353119 
.352778 
.352433 
.352097 
.351757 
.351417 


M. 

GO 
.59 
53 
57 
56 
55 
54 
53 
52 
51 

50 

49 

48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
23 
27 
26 
2-5 
24 
23 
22 
21 

20 
19 
13 
17 
16 
15 
14 
13 
12 
11 

10 

9 


6 
5 
4 
3 
2 
1 
0 


Tang.   M. 


(QQC 


198 


TABLE    Xlll.       LOGARITHMIC    SINES, 


155<. 


M. 


M. 


isine. 


0 

9.609313 

I 

.609597 

2 

.609S80 

3 

.610164 

4 

.61/)147 

5 

.610729 

6 

.611012 

7 

.611294 

8 

.611576 

9 

.611853 

10 

9.612140 

11 

.612421 

12 

.612702 

13 

.612933 

14 

.613264 

15 

.61.3545 

16 

.613325 

17 

.614105 

13 

.614385 

19 

.614665 

2D 

9.614944 

21 

.615223 

22 

.615.502 

23 

.615731 

24 

.616030 

25 

.616333 

26 

.616516 

27 

.616394 

23 

.617172 

29 

.617450 

30 

9.617727 

31 

.618004 

32 

.618231 

33 

.618.553 

34 

.613334 

35 

.619110 

36 

.619336 

37 

.619662 

38 

.619933 

39 

.620213 

40 

9.620133 

41 

.620763 

42 

.621033 

43 

.621313 

44 

.621537 

45 

.621861 

46 

.6221.35 

47 

.622409 

48 

.622632 

49 

.622956 

50 

9.623229 

51 

.623502 

52 

.6^3774 

53 

.624047 

54 

.624319 

55 

.624591 

56 

.624363 

57 

.625135 

58 

.625406 

59 

.625677 

60 

.625948 

D.  1". 


Cosine. 


4.73 
4.72 
4.72 
4.72 
4.71 
4.71 
4.71 
4.70 
4.70 
4.69 

4.69 
4.69 
4.63 
4.63 
4.63 
4.67 
4.67 
4.67 
4.66 
4.66 

4.65 
4.65 
4.65 
4.64 
4.61 
4.64 
4.63 
4.63 
4.63 
4.62 

4.62 
4.61 
4.61 
4.61 
4.60 
4.60 
4.60 
4.59 
4.59 
4.59 

4.53 
4.53 
4.53 
4.57 
4..57 
4..57 
4.56 
4.56 
4.56 
4.55 

4.55 
4.54 
4.54 
4.. 54 
4.53 
4.53 
4.53 
4.52 
4..52 
4,52 

D.  1". 


Cosine. 


D.  1". 


9.9607.30 
.960674 
.960618 
.96)561 
.960505 
.960443 
.960392 
.960335 
.96)279 
.960222 

9. 96  T 165 
.960109 
.96)052 
.959995 
.959933 
.9.59332 
.959325 
.9.59768 
.959711 
.959654 

9  959596 
,959539 
,959432 
,959425 
,959363 
,959310 
.9.59253 
.959195 
.9.59133 
.959030 

9.959023 
.953965 
.958908 
.9533.50 
.953792 
.9587.34 
.953677 
.953619 
.953.561 
.953503 

9.958445 
.953337 
.953329 
.953271 
.9.53213 
.9.53154 
.9.58096 
.958038 
.957979 
.957921 

9.957863 
.957804 
.957746 
.957637 
.957623 
,957570 
,957511 
.957452 
.957393 
.9573a5 
,957276 

Sine. 


,94 
,94 

.94 
,94 
,94 
.94 
.94 
.94 
.94 
.94 

.95 
.95 
.95 
.95 
,95 
.95 
.95 
.95 
,95 
.95 

.95 
.95 
.95 
.95 
.96 
.96 
,96 
.96 
,96 
,96 

.96 
.96 
.98 
.96 
.96 
.96 
.96 
.97 
.97 
,97 

.97 
.97 
.97 
.97 
.97 
.97 
,97 
.97 
,97 
.97 

.97 
.93 
.93 
.93 
,98 
.98 
.93 
.93 
,98 
.98 

D.  1". 


Tang, 


9.648583 
.643923 
.649263 
.649002 
.649942 
.650231 
.650620 
.650959 
.651297 
.651636 

9.651974 
.652312 
.652650 
.652933 
.653326 
.653663 
.6.54000 
,654337 
.654674 
.655011 

9.655343 
.65.5634 
.656020 
.656356 
.656692 
.6.57023 
.657364 
.657699 
,653034 
,653369 

9.653704 
.659039 
,6.59373 
.6-59703 
.660042 
.660376 
.660710 
.661043 
.661-377 
.661710 

9.662043 
.662376 
.662709 
.663042 
.663375 
.663707 
,664039 
.664371 
.664703 
.665035 

9.66-5366 
.66.5693 
.666029 
.666360 
,666691 
,667021 
.6673^52 
.667682 
.663013 
.663343 
,668673 

Cotang. 


D.  1'. 


5.67 
5.66 
5.66 
5.66 
5.65 
5.65 
5.65 
5.64 
5.64 
5.64 

5.64 
5.63 
5.63 
5.63 
5.62 
5.62 
5.62 
5.62 
5.61 
5.61 

5.61 
5.61 
5.60 
5.60 
5.60 
5.  .59 
5.59 
5.59 
5.58 
5.58 

5.-58 
5.58 
5.57 
5.57 
5.57 
5.56 
5.56 
5.56 
5.. 56 
5.55 

5.55 
5.55 
5.54 
5.54 
5.54 
5.54 
5.53 
5.-53 
5.-53 
5.53 

5.52 
5.52 
5,  .52 
5.51 
5.51 
5.51 
5.51 
5.50 
5.50 
5.50 

D.  1". 


Cotang, 


0.351417 
.351077 
.350737 
.350398 
.350058 
.349719 
.349380 
.349041 
,343703 
.348364 

0.343026 
.347638 
.347a50 
.347012 
.346674 
.346337 
.346000 
.345663 
,345326 
.344939 

0.344652 
.344316 
,34.3930 
,343644 
,313308 
,342972 
.31^636 
.342301 
.341966 
.341631 

0.3-11296 
.340961 
..340627 
,340292 
.3-39953 
,a39624 
,339290 
.333957 
,a33623 
,333290 

0.337957 
,.337624 
.337291 
.336958 
,3-36625 
.336293 
.3.35961 
.335629 
,3-35297 
.33496-. 

0.3-34634 
.334302 
.333971 
.-333640 
.-3-33309 
.332979 
.332643 
.332318 
.331987 
.3316.57 
,331327 

Tang. 


1140 


690 


COSINES,    TANGENTS,    AND    COTANGENTS. 


199 

154ta 


M.        Sine. 


D.  1". 


0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 

14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

M. 


9.625948 
.626219 
.626490 
.62(3760 
.627030 
.627300 
.627570 
.627840 
.62S109 
.628378 

9.628647 
.623916 
.629185 
,629453 
.620721 
.629989 
.630257 
.630524 
.630792 
.631059 

9.631326 
.631593 
.631859 
.632125 
.632392 
.6326.58 
.632923 
.633189 
.633454 
.633719 

9.633934 
.634249 
.634514 
.634778 
.635042 
.635306 
.635570 
.635834 
.636097 
.636360 

9.636623 
.636886 
.637148 
.637411 
.637673 
.637935 
.638197 
.638458 
.638720 
.63S081 

9.639242 
.639503 
.639764 
.610f/24 
.640284 
.frl0544 
.640304 
.641064 
.641324 
.&11583 
.641842 

Cosine. 


4.51 

4.51 

4.51 

4.50 

4.50 

4.50 

4.49 

4.49 

4.49 

4.48 

4.48 
4.48 
4.47 
4.47 
4.47 
4.46 
4.46 
4.46 
4.45 
4.45 

4.45 
4.44 
4.44 
4.44 
4.43 
4.43 
4.43 
4.42 
4.42 
4.42 

4.41 

4.41 

4.41 

4.40 

4.40 

4.40 

4.39 

4.39 

4.39 

4.33 

4.38 
4.38 
4.37 
4.37 
4.37 
4.36 
4.36 
4.36 
4.35 
4.35 

4.35 
4.34 
4.34 
4.34 
4.33 
4.33 
433 
4.32 
4.32 
4.32 

D.  1". 


Cosine. 

9.957276 
.957217 
.957158 
.957099 
.957040 
.956981 
.956921 
.956^62 
.956S03 
.956744 

9. 9566.84 
.956625 
.956566 
.956506 
.956447 
.956.387 
.956327 
.956268 
.956208 
.956148 

9.956089 
.956029 
.955969 
.95.5909 
.955849 
.955789 
.955729 
.955669 
.955609 
.955548 

9.955488 
.955428 
.955363 
.955307 
.95.5247 
.955186 
.955126 
.955065 
.955005 
.954944 

9.954883 
.954823 
.954762 
.954701 
.954640 
.954.579 
.9.54518 
.954457 
.954396 
.954335 

9.954274 
.954213 
.954152 
.954090 
.954029 
.953968 
.953906 
.953845 
.953783 
.953722 
.953660 

Sine. 


D  1". 


Tang. 


D.  1". 


.98 
.98 
.98 
.98 
.99 
.99 
.99 
.99 
.99 
.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 

.99 
1.00 
1.00 

1.00 
1.00 

1. 00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 

1.00 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 
1.01 

1.01 
1.01 
1.01 
1.01 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 

1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.03 
1.03 
1.03 

D.  1". 


9.66-:673 
.669002 
.669332 
.669661 
.669991 
.670320 
.670649 
.670977 
.671306 
.671635 

9.671963 
.672291 
.672619 
.672947 
.673274 
.673602 
.673929 
.674257 
.674.584 
.674911 

9.675237 
.675564 
.675890 
.676217 
.676543 
.676869 
.677194 
.677520 
.677846 
.678171 

9.678496 

.678821 
.679146 
.679471 
.679795 
.680120 
.680444 
.680768 
.681092 
.681416 

9.681740 
.682063 
.682387 
.682710 
.683033 
.683356 
.683679 
.684001 
.684324 
.634646 

9.684968 
.685290 
.68.5612 
.6S5934 
.686255 
.686577 
.686898 
.687219 
687540 
687861 
.688182 

Gotang. 


Cotang. 


5.50 
5.49 
5.49 
5.49 
5.49 
5.48 
5.48 
5.48 
5.47 
5.47 

5.47 
5.47 
5.46 
5.46 
6.46 
5.46 
5.45 
5.45 
5.45 
5.45 

5.44 
5.44 
5.44 
5.44 
5.43 
5.43 
5.43 
5.42 
5.42 
5.42 

5.42 
5.41 
5.41 
5.41 
5.41 
5.40 
5.40 
5.40 
5.40 
5.39 

5.39 
5.39 
5.39 
5.38 
5.38 
5.38 
5.33 
5.37 
6.37 
6.37 

5.37 
6.36 
5.36 
5.36 
5.36 
5.35 
6.35 
5.35 
6.35 
5.35 

J).V. 


M. 


0.331327 
.330998 
.330668 
.330339 
.330009 
.3296^0 
.329351 
.329023 
.328694 
.328365 

0.328037 
.327709 
.327381 
.327053 
.326726 
.326398 
.326071 
.325743 
.325416 
.325089 

0.324763 
.324436 
.324110 
.323783 
.323457 
.323131 
.322806 
.322480 
.322154 
321829 

0.321504 
.321179 
.320854 
.320529 
.320205 
.319880 
.319556 
.319232 
.318908 
.318584 

0.318260 
.317937 
.317613 
.317290 
.316967 
.316644 
.316321 
.315999 
,315676 
.315354 

0.315032 
.314710 
.314388 
.314066 
.313745 
.313423 
.313102 
.312781 
,312460 
.312139 
.311818 


Tuc. 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 

M. 


1150 


640 


200 


TABLE     Xlll.        LOGAKITHMIC    SINES, 


153> 


M. 


0 
1 
2 
3 

4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
IS 
19 

20 
21 
22 
23 
21 
2.-, 
26 
27 
2S 
29 

.30 
31 
82 
33 
31 
35 
36 
37 
33 
39 

40 
41 

42 
43 
44 
45 

46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


9.641812 
.642101 
.642360 
.64 26  IS 
.642877 
.643135 
.613393 
.6136.50 
.643908 
.644165 

9.641123 
.614'J80 
.644,(36 
.645193 
.615150 
.645700 
.64-5962 
.646218 
.646474 
.646729 

9.646984 
.647210 
.647194 
.647749 
.648004 
.64S258 
.648512 
.643766 
.649020 
.649274 

9.649527 
.649781 
.650:)34 
.65y2s7 
.650.539 
.650792 
.651044 
.651297 
.651549 
.651800 

9.6.520.52 
.652304 
.652555 
.652806 
.653057 
.653.303 
.653553 
.653303 
.6.54059 
.654309 

9.654553 
.6.54303 
.6.550-58 
.6.55307 
.655556 
.655805 
.656054 
.656302 
.656551 
.656799 
.657047 


D.  1". 


4.32 
4.31 
4.31 
4.31 
4.30 
4.30 
4.30 
4  29 
4.29 
4.29 

4.28 
4.28 
4.23 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.26 

4.25 
4.25 
4.25 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 
4.22 

4.22 
4.22 
4.22 
4^21 
4.21 
4.21 
4.20 
4.20 
4.20 
4.19 

4.19 
4.19 
4.18 
4.18 
4.13 
4.18 
4.17 
4.17 
4.17 
4.16 

4.16 
4.16 
4.15 
4.15 
4.15 
4.15 
4.14 
4.14 
4.14 
4.13 


M. 


1163 


Cosine.   D.  1" 


Cosine. 


9.953660 
.953599 
.953537 
.95.3475 
.9.53113 
.953352 
.953290 
.953228 
.9.53166 
.953104 

9.9.53042 
.9.52980 
.9-52918 
.952855 
.952793 
.9.52731 
.952669 
.9.52606 
.9-52.544 
.952481 

9.952419 
.952356 
.952234 
.952231 
.952168 
.952106 
.952043 
.951980 
.951917 
.9518-54 

9.951791 
.951723 
.951665 
.951602 
.951539 
.951476 
.951412 
.951319 
.9.J1286 
.951222 

9.951159 
.951096 
.9510-32 
.9-50963 
.950905 
.9-50841 
.9-50778 
.950714 
.9506-50 
.950586 

9.950522 
.950453 
.950394 
.9.50330 
.950266 
.950202 
.9-50133 
.9.50074 
.9.50010 
.949945 
.949381 


Sine. 


D.  1". 


1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
1.03 
I  03 
1.03 
1.03 

1.03 
1.04 
1.01 
1.04 
1.04 
1.04 
1.04 
1.01 
1.04 
1.04 

1.04 
1.04 
1.04 
1.01 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 

1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.05 
1.06 
1.06 
1.06 

1.06 
1.06 
1.06 
1.06 
1.06 
1.06 
1.06 
1.06 
1.06 
1.06 

1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 


D.  1". 


Tang. 


9.688182 
.638502 
.633823 
.639143 
.639463 
.639783 
.690103 
.690423 
.69  1742 
.6J1062 

9.691-381 
.6  J 1700 
.692019 
.692333 
.692656 
.692975 
.693293 
.693612 
.693930 
.694248 

9.694566 
.694833 
.69-5201 
.695518 
.69-5336 
.6961.53 
.696470 
.696787 
.697103 
.697420 

9.697736 
.6:)8053 
.693369 
.693635 
.699001 
.699316 
.699632 
.699947 
.700263 
.700578 

9.700393 
.701208 
.701.523 

.7018.37 
.7021.52 
.702466 
.702781 
.703095 
.7034.^9 
.703722 

9.704036 
.7043-50 
.704663 
.704976 
.705290 
.70560-3 
.70.5916 
.706228 
.706541 
.7063-54 
.707166 


Cotang. 


D.  1". 


5.34 
5.34 
5.34 
5.34 
5.-33 
5.33 
5.33 
5.33 
5.32 
5.32 

5.32 
5.-32 
5.31 
5.31 
5.31 
5.31 
5.30 
5.-30 
.5.. 30 
5.30 

5.29 
5.29 
5.29 
5.29 
5.29 
5.23 
5.28 
5.23 


5.27 

5.27 
5.27 
5.27 
5.26 
5.26 
5.26 
5.26 
5.26 
5.25 
5.25 

5.25 
5.25 
5.24 

5.24 


5.24 
.5.24 
5.23 
5.23 
5.23 

5.23 
5.22 
5.22 
5.22 
5.22 
5.22 
5.21 
5  21 
5.21 
5.21 


D.  1". 


Cotang. 

M. 

0.311813 

60 

.311498 

59 

.311177 

58 

.310S57 

57 

.310-5.37 

56 

.310217 

55 

.309397 

54 

..309577 

53 

J30925S 

52 

.308938 

51 

0..305619 

50 

.308300 

49 

.307981 

48 

.307662 

47 

.307344 

46 

.307025 

45 

..306707 

44 

.306338 

43 

.306070 

42 

.305752 

41 

0.30.5434 

40 

.305117 

39 

.304799 

33 

.304482 

37 

.301164 

36 

.303347 

35 

303530 

34 

.303213 

33 

.3  12397 

32 

.302580 

31 

0.302264 

30 

.301947 

29 

.3)1631 

28 

.301315 

27 

.30f)999 

26 

..3011634 

25 

.300363 

24 

.300053 

23 

.299737 

22 

.299422 

21 

0.299107 

20 

.298792 

19 

.298477 

18 

.293163 

17 

.297848 

16 

.297534 

15 

.297219 

14 

.296905 

13 

.296.591 

12 

.296278 

11 

0.295964 

10 

.295650 

9 

.295337 

8 

.295r.<24 

7 

.294710 

6 

.294-397 

5 

.294034 

4 

.293772 

3 

.293459 

2 

.293146 

1 

.2923.34 

0 
M. 

Tang. 

e3< 


COSINES,    TANGENTS,    AND    COTANGENTS. 


201 

153- 


M. 


-I- 


Siiie. 


10 
II 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
2.3 
26 
27 
23 
29 

30 
31 
32 
33 

36 

37 
33 
39 

10 
II 
12 
13 
14 
15 
16 
17 
IS 
19 

50 
51 
52 
53 
54 
55 
56 
57 
5S 
5£ 
6C 


D.  1". 


9.637017 
.657293 
.637542 
.657790 
.653037 
.6.58234 
.6.53531 
.653773 
639025 
.659271 

G  659317 
.639763 
.660009 
.660255 
.660501 
.660746 
.660991 
.66123G 
.661-131 
.6t)1726 

9.661970 
.6;2214 
.6521.59 
.662703 
.662916 
.663190 
.663433 
.663677 
.663920 
.664163 

9.661106 
.664643 
.661391 
.665133 
.665375 
.665617 
.66.5859 
.666100 
.666342 
.666533 

9.666324 
.667065 
,667305 
.667346 
.667736 
.663027 
.663267 
.663506 
.663746 
.663936 

9.669225 
.669464 
.669703 
.669942 
.670131 
.670419 
,670653 
.670396 
.6711.34 
.671372 
.671609 


4.13 
4.13 
4.12 
4.12 
4.12 
4.12 
4.11 
4.11 
4.11 
4.10 

4.10 
4.10 
4.10 
4.09 
4.09 
4.09 
4.03 
4.03 
4.03 
4.03 

4.07 
4.07 

4.07 
4.05 
4.06 
4.06 
4.05 
4.03 
4.03 
4.05 

4.04 
4.04 
4.04 
4.03 
4.03 
4.03 
4.03 
4.02 
4.02 
4.02 

4.01 
4.01 
4.01 
4.01 
4.00 
4.00 
4.00 
3.99 
3.99 
399 

3.99 
3.93 
3.9? 
3.93 
3.93 
3.97 
3.97 
3.97 
3.96 
3.96 


Cosiue. 


D.  1" 


9.919331 
.919316 
.949752 
.919658 
.949623 
.949353 
.949194 
.919429 
.949364 
.949300 

9.949235 
.949170 
.949105 
.949040 
.948975 
.943910 
.913345 
,943730 
.943715 
,913650 

9.943.531 
.913319 
.943454 
.943338 
.943323 
,913257 
,918192 
.943126 
.943060 
.947995 

9.947929 
.947863 
.947797 
.947731 
.947665 
.917600 
.947533 
.917467 
.917401 
.947333 

9.947269 
.947203 
.947136 
.947070 
.9170r)l 
.916937 
.946371 
.946304 
.946733 
,916671 

9,916604 
.946533 
.946471 
.946404 
.946337 
.916270 
,946203 
.946136 
.916069 
.916(02 
.9439.35 


1.07 
1.07 
1,07 
1,03 
1.03 
1.03 
1.03 
1.03 
1.03 
1.03 

1.03 
1.03 
1.03 
1.03 
1,03 
1.08 
1,09 
1.09 
1.09 
1.09 

1.09 
1,09 
1,09 
1,09 
1,09 
1,09 
1,09 
1.09 
1.09 
1.10 

1.10 
1.10 
1.10 
1. 10 
1.10 
1.10 
1. 10 
l.IO 
1. 10 
1. 10 

l.!0 
1.11 
1,11 
l.ll 
1.11 
1,11 
1.11 
1.11 
l.U 
1.11 

1.11 
l.U 
1.11 
1.11 
1.12 
1.12 
1.12 
1.12 
1.12 
1  12 


Tang. 


9.707166 
.707478 
.707790 
.703102 
.703414 
.703726 
.709037 
.709349 
.709660 
.709971 

9.710232 
.710593 
.710304 
.711215 
,71 1525 
.711336 
.712146 
.712156 
.712766 
,713076 

9,71.3336 
,713696 
,714005 
,714314 
,714624 
,714933 
,71.5242 
.715551 
.715360 
.716168 

9,716477 
,716735 
.717093 
.717401 
,717709 
,713017 
.718.325 
,718633 
.713940 
.719213 

9.719.555 
.719362 
.720169 
.720176 
.720783 
.721039 
,721396 
.721702 
,722009 
,722315 

9,722621 
.722927 
.723232 
.723338 
,723344 
,724149 
.7244.54 
.724760 
.725065 
.723370 
.725674 


M.   Cosine,   D.  1",    Sine,    D,  1",   Cotang,  D,  1" 


D.  1". 


5.20 
5.20 
5.20 
5,20 
.5.20 
5,19 
5.19 
5.19 
5.19 


5,13 
5.18 
5.18 
5.13 
5,17 
5,17 
5.17 
5.17 
.5,17 
5,16 

5,16 
5,16 
5.16 
5.15 
5.15 
5.15 
5.15 
5.15 
5.14 
5.14 

5.14 
5.14 
5.14 
5.13 
5,13 
5,13 
5.13 
5.13 
5.12 
5.12 

5.12 
5.12 
5.11 
5.11 
5.11 
5.11 
5.11 
5.10 
5.10 
5.10 

5.10 
5.10 
5.09 
5.09 
5.09 
5.09 
5.09 
5,08 
5,08 
5.03 


Cotang.   M 


0,292834 
,292522 
,292210 
.291893 
,291536 
,291274 
,290963 
.290651 
.290340 
,290029 


60 
59 
58 
57 
56 
53 
54 
53 
52 
51 


0.239718  . 

50 

.239407 

49 

.239096 

48 

.238785 

47 

.238475 

46 

.233164 

45 

.237854 

44 

.237514 

43 

.2372.34 

42 

.236924 

41 

0.2S6614 

40 

.236304 

39 

.235995 

33 

.235836 

37 

.285376 

36 

.235067 

35 

.234753 

34 

.234149 

33 

.281140 

32 

.233332 

31 

0.283523 

30 

.233215 

29 

.232907 

23 

.232.599 

27 

.232291 

26 

.231933 

23 

.231675 

24 

.231367 

23 

.231060 

22 

.230752 

21 

0.280445 

20  ' 

.2301.33 

19 

.279331 

13 

,279324 

17 

,279217 

16 

.273911 

15 

,273604 

14 

.273293 

13 

,277991 

12 

.277635 

11 

0,277379 

10 

,277073 

9 

.276768 

8 

,276462 

7 

.276156 

6 

.275351 

5 

.275.546 

4 

.275240 

3 

.274935 

2 

.274630 

1 

.274326 

0 
M. 

Tang. 

1170 


10 


6a« 


202 

280 


TABLE    XIII.       LOGARITHMIC    SINES, 


131 


M. 


0 
I 

2 
3 

4 
5 
6 

7 
8 
9 

10 
II 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine. 


9.671609 
.671847 
.672034 
.672321 
.672553 
.672795 
.673032 
.673263 
.673505 
.673741 

9.673977 
.674213 
.674443 
.674634 
.674919 
.675155 
.675390 
.67.5624 
.67.5359 
.676094 

9.676323 
.676.562 
.676796 
.677030 
.6772&4 
.677493 
.677731 
.677964 
.678197 
.673430 

9.673663 
.673395 
.679123 
.679360 
.679.592 
.679324 
.630056 
.630233 
.630519 
.630750 

9.630932 
.631213 
.631443 
.631674 
.631905 
.6321.35 
.632365 
.632595 
.632325 
.6S3055 

9.633234 
.633514 
.633743 
.633972 
.634201 
.634430 
.634653 
.634337 
.63.S115 
.63.5343 
.635571 


M.      Cosine. 


D.  1". 


3.96 
3.96 
3.95 
3.95 
3,95 
3  94 
3,94 
3.94 
3.94 
3.93 

3.93 
3.93 
3.93 
3.92 
3.92 
3.92 
3.91 
3.91 
3.91 
3.91 

3.90 
3.90 
3.90 
3.90 

3. 39 
3.89 
3.89 
3.83 
3.83 
3.33 

3.83 
3.87 
3.87 
3.37 
3.37 
3.86 
3.86 
3.86 
3.86 
3.85 

3.85 
3.35 
3.34 
3.84 
3.84 
3.84 
3.83 
3.33 
3.33 
3.83 

3.32 
3.82 
3.82 
3.82 
3.81 
3.81 
3.81 
3.80 
3.80 
3.80 


D.  1". 


Cosine. 


9.94.5935 

.945363 
.945800 
.945733 
.945666 
.945593 
.94.5.531 
.94.5464 
.945396 
.945323 

9.94.5261 
.945193 
.945125 
.945053 
.944990 
.944922 
.944354 
.944786 
.944718 
.9446.50 

9.944.532 
.944514 
.944446 
.944377 
.944.309 
.944241 
.944172 
.944104 
.944036 
.943967 

9.943S99 
.9433.30 
.943761 
.943693 
.943624 
.943555 
.943436 
.943417 
.94.3343 
.94.3279 

9.943210 
.943141 
.94.3072 
.943003 
.9429-34 
.942364 
.942795 
.942726 
.942656 
.942.587 

9.942517 
.942443 
.942373 
.942.308 
.942239 
.942169 
.942099 
.942029 
.941959 
.941839 
.941819 


Sine. 


D.  1". 


,12 
,12 
,12 
,12 
,12 
,12 
,12 
,13 
,13 
,13 

,13 
,13 
,13 
,13 
,13 
,13 
,13 
,13 
,13 
,13 

,14 
,14 
,14 
,14 
,14 
14 
,14 
,14 
14 
14 

14 
14 
15 
15 
15 
15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
17 
17 
17 


D.  1". 


Tang. 


9.725674 
.725979 
.726234 
.726538 
.726392 
./27197 
.727.501 
.727805 
.723109 
.723412 

9.723716 
.729020 
.729.323 
.729626 
.729929 
.730233 
.730535 
.730338 
.731141 
.731444 

9.731746 
.732043 
.732351 
.732653 
.732955 
.733257 
.733558 
.733360 
.734162 
.734463 

9.734764 
.735066 
.735367 
.735663 
.735969 
.736269 
.736570 
.786370 
,737171 
.737471 

9.737771 
.733071 
.733371 
.733671 
.735971 
.739271 
.739570 
.739370 
.740169 
.740468 

9.740767 
.741066 
.741365 
.741664 
.741962 
.742261 
.742559 
.742858 
.743156 
.743454 
.743752 


D,  1". 


Cotang. 


5.08 
5.08 
5.07 
5.07 
5.07 
5.07 
5.07 
5.06 
5.06 
5.06 

5.06 
5.06 
5.05 
5.05 
5.05 
5.05 
5.05 
5.05 
5.04 
5.04 

5.04 
5.04 
5.04 
5.03 
5.03 
5.03 
5.03 
5.03 
5.02 
5.02 

5.02 
5.02 
5.02 
5.01 
5.01 
5.01 
5.01 
5.01 
5.01 
5.00 

5.00 
5.00 
5.00 
5.00 
4.99 
4.99 
4.99 
4.99 
4.99 
4.93 

4.98 
4.98 
4.93 
4.93 
4.93 
4.97 
4.97 
4.97 
4.97 
4.97 


D.  1". 


Cotang. 

0.274326 
.274021 
.273716 
.273412 
.273103 
.272303 
.272499 
.272195 
.271891 
.271588 

0.271234 
.270930 
.270677 
.270374 
.270071 
.269767 
.269465 
.269162 
.263859 
.268556 

0.2632.54 
.267952 
.267649 
.267347 
.267045 
.266743 
.266442 
.266140 
.265833 
.265537 

0.2652.36 
.264934 
.261633 
.264-332 
.264031 
.263731 
.263430 
.263130 
.262,329 
.262529 

0.262229 
.261929 
.261629 
.261329 
.261029 
.260729 
.260430 
.260130 
.2.59331 
.259532 

0.259233 
.253934 
.2.58635 
.258336 
.253033 
.257739 
.257441 
.257142 
.2.56344 
.2.56.546 
.256243 


Tang. 


118' 


COSINES,    TANGENTS,    AND    COTANGENTS. 


M 


Sine. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

!0 
11 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


D.  1". 


9.635371 
.6S579'J 
.6S6027 
.656254 
.636432 
.636709 
.656936 
.637163 
.637339 
.637616 

9.657313 
.653069 
.633295 
.655521 
.633747 
.655972 
.659193 
.6>9123 
.6^9615 
.659573 

9.690095 
.890323 
.690543 
.693772 
.690996 
.691220 
.691444 
.691665 
.691892 
.692115 

9.692339 
.692562 
.692785 
.693003 
.693231 
.693453 
.693676 
.693593 
.694120 
.694342 

9.694.564 
.694736 
.69.5007 
.695229 
.695450 
.69.5671 
.695392 
.696113 
.696334 
.696554 

9.696775 
.696995 
.697215 
.697435 
.697654 
.697874 
.695094 
.698313 
.698532 
.693751 
.698970 


Cosine. 


D.  1". 


M.  Cosine. 


3.80 
3.79 
3.79 
3.79 
3.79 
3.78 
3.78 
3.78 
3.73 
3.77 

3.77 
3.77 
3.77 
3.76 
3.76 
3.76 
3.76 
3.75 
3.75 
3.75 

3.75 
3.74 
3.74 
3.74 
3.74 
3.73 
3.73 
3.73 
3.73 
3.72 

3.72 
3.72 
3.72 
3.71 
3.71 
3.71 
3.71 
3.70 
3.70 
3.70 

3.70 
3.69 
3.69 
3.69 
3.69 
3.63 
3.63 
3.63 
3.63 
3.67 

3.67 
3.67 
3.67 
3.66 
3.66 
3.66 
3.66 
3.65 
3.65 
3.65 


9.941819 
.911749 
.941679 
.911609 
.941539 
.941469 
.941393 
.941323 
.941253 
.941137 

9.941117 
.941046 
.910975 
,940905 
.940334 
.940763 
.940693 
.940622 
.940551 
.940480 

9.940409 
.940333 
.940267 
.940196 
.940125 
.940054 
.939982 
.939911 
.939340 
.939768 

9.939697 
.939625 
.9.395.54 
.939482 
.939410 
.939339 
.939267 
.939195 
.939123 
.939052 

9.935930 
.935908 
.933336 
.933763 
.933691 
.935619 
.933.547 

■ .933475 
.933402 
.933330 

9.9382.58 
.933185 
.933113 
.933040 
.937967 
.937895 
.937822 
.937749 
.937676 
.937604 
.937531 


Tang. 


D.  1". 


Sine. 


1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 
1.17 

1.18 
1.18 

1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 
1.18 

1.18 
1.18 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 
1.19 

1.19 
1.19 
1.19 
1.19 
1.19 
1.20 
1.20 
1.20 
1.20 
1.20 

1.20 
1.20 
1.20 
1.20 
1  20 
1.20 
1.20 
1.21 
1.21 
1.21 

1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.21 
1.22 


D.  1". 


9.743752 
.744050 
.744343 
.744645 
.744943 
.745240 
.74.5533 
.74.5335 
.746132 
.746429 

£.  r46726 
.747023 
.747319 
.747616 
.747913 
.748209 
.748505 
.748801 
.749097 
.749393 

9.749639 
.749935 
.750281 
.750576 
.750872 
.751167 
.751462 
.751757 
.752052 
.752347 

9.752642 
.7.52937 
.753231 
.753526 
.753320 
.754115 
.754409 
.754703 
.754997 
.755291 

9.7.5.5585 
.755373 
.756172 
.756165 
.756759 
.757052 
.757345 
.757633 
.7.57931 
.758224 

9.753517 
.758810 
.759102 
.759395 
.759687 
.759979 
.760272 
.760564 
.760856 
.761148 
.761439 


Cotang. 


D.  1".  I  Cotang. 


4.96 
4.96 
4.96 
4.96 
4.96 
4.96 
4.95 
4.95 
4.95 
4.95 

4.95 
4.95 
4.94 
4.94 
4.94 
4.94 
4.94 
4.93 
4.93 
4.93 

4.93 
4.93 
4.93 
4.92 
4.92 
4.92 
4.92 
4.92 
4.92 
4.91 

4.91 
4.91 
4.91 
4.91 
4.91 
4.90 
4.90 
4.90 
4.90 
4.90 

4.89 
4.89 
4.89 
4.89 
4.89 
4.89 
4.88 
4.83 
4.83 
4.88 

4.88 
4.88 
4.87 
4.87 
4.87 
4.87 
4.87 
4.87 
4.86 
4.86 


0.256243 
.255950 
.255652 
.255355 
.255057 
.254760 
.25-1462 
.2.54165 
.253363 
.253571 

0.253274 
.252977 
.2.52031 
.252334 
.252037 
.251791 
.251495 
.251199 
.250903 
.250607 

0.2.50311 
.250015 
.249719 
.249424 
.249123 
.243833 
.248538 
.248243 
.247943 
.247653 

0.247358 
.247063 
.246769 
.246474 
.246180 
.24.5835 
.245591 
.245297 
.24.5003 
.244709 

0.244415 
.244122 
.243325 
.243535 
.243241 
.242948 
.242655 
.242362 
.242069 
.241776 

0.241483 
.241190 
.240398 
.240605 
.240313 
.240021 
.239723 
.239436 
.2.39144 
.238852 
.238561 


D.  1".   Tang. 


1190 


60< 


204 

30^ 


TABLE     ^'III.       LOGARITHMIC    SINES, 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
II 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
2:5 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 

42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


D.  1". 


9.693970 
.699139 
.699407 
.699626 
.699844 
.700062 
.700230 
.700493 
.700716 
.700933 

9.701151 
.701363 
.701.585 
.701302 
.702019 
.702236 
.702452 
.702669 
.702335 
.703101 

9.703317 
.7()3533 
.703749 
.703664 
.704179 
.704395 
.704610 
.704S25 
.705040 
.705254 

9.705469 
.705633 
.705398 
.706112 
.706326 
.706.539 
.706753 
.706967 
.707130 
.707393 

9.707606 
.707819 
.703032 
.703245 
.7034.33 
.703670 
.703832 
.709094 
.709306 
.709518 

9.709730 
.709941 
.710153 
.710364 
.710575 
.710736 
.710997 
.711208 
.711419 
.711629 
.711839 


3.65 
3.64 
3.64 
3.64 
3.64 
3.63 
3.63 
3.63 
3.63 
3.62 

3.62 
3.62 
3.62 
3.61 
3.61 
3.61 
3.61 
3.60 
3.60 
3.60 

3.60 
3.59 
3.59 
3.59 
3.  .59 
3.59 
3.58 
3.  .58 
3.53 
3.. 53 

3.57 
3.57 
3.. 57 
3.57 
3.-56 
3.56 
3.56 
3.56 
3.55 
3.55 

3.55 
3.55 
3.54 
3.54 
3.54 
3.54 
3.54 
3.53 
3.53 
3.53 

3.53 
3.52 
3.52 
3.52 
3.52 
3.51 
3.51 
3.51 
3.51 
3.51 


Cosine.   D,  1". 


Cosine. 

9.937531 
.937453 
.937.335 
.937312 
.937233 
.937165 
.937092 
.937019 
.936946 
.936872 

9.9.36799 
.936725 
.936652 
.936578 
.936.505 
.936431 
.936-357 
.936284 
.936210 
.936136 

9.936062 
.935938 
.935914 
.9-3-5340 
.935766 
.935692 
.935618 
.9-35543 
.935469 
.935395 

9.93.5320 
.935246 
.9-35171 
.935097 
.935022 
.934943 
.934873 
.934793 
.9-34723 
.934649 

9.934574 
.9.34499 
.934424 
.934349 
.934274 
.934199 
.9-34123 
.934043 
.93-3973 
.933S98 

9.93-3822 
.933747 
.933671 
.933596 
933520 
933445 
933369 
933293 
933217 
.933141 
.933066 


D.  1". 


Sine. 


1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 
1.22 

1.22 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 
1.23 

1.23 

1.23 
1.23 
1.23 
1.24 
1.24 
1.24 
1.24 
1.24 
1.24 

1.24 
1.24 
1.24 
1.24 
1.24 
1.24 
1.25 
1.25 
1.25 
!.!.5 

1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
1.25 
I.2G 
1.26 

1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 
1.26 


Tang. 


D.  1". 


9.761439 
.761731 
.762023 
.762314 
.762606 
.762897 
.763133 
.76.3479 
.763770 
.764061 

9.764.352 
.764643 
.764933 
.765224 
.765514 
.765305 
.766095 
.766385 
.766675 
.766965 

9.767255 
.767.545 
.7678.34 
.768124 
.768414 
.768703 
.763992 
.769231 
.769-571 
.769560 

9.770148 
.770437 
.770726 
.771015 
.771.303 
.771592 
.771830 
.772163 
.772457 
.772745 

9,7730-33 
.773-321 
.773608 
.773896 
.774184 
.774471 
.774759 
.775046 
.7753-33 
.775621 

9.775908 
.776195 
.776482 
.776763 
.777055 
.777342 
.777623 
.777915 
.773201 
.773488 
.778774 


D.  1". 


Cotang. 


Cotang. 


4.S6 
4.86 
4.S6 
4.86 
4.S6 
4.85 
4.85 
4.85 
4.85 
4.85 

4. 35 
4.84 
4.84 
4.84 
4.84 
4.84 
4.84 
4.83 
4.83 
4.83 

4.83 
4.83 
4.83 

4.82 
4.32 
4.82 
4.82 
4.82 
4.82 
4.82 

4.81 
4.81 
4.81 
4.81 
4.SI 
4.81 
4.80 
4.80 
4.S0 
4.80 

4.80 
4.80 
4. SO 
4.79 
4.79 
4.79 
4.79 
4.79 
4.79 
4.78 

4.78 
4.78 
4.78 
4.78 
4.78 
4.78 
4.77 
4.77 
4.77 
4.77 

D.  1". 


0.238561 
.238269 
.237977 
.237686 
.237394 
.237103 
.236312 
.2-36521 
.236230 
.235939 

0.235643 
.235357 
.235067 
.234776 
.234486 
.234195 
.2339C5 
.233615 
.233325 
.233035 

0.232745 
.232455 
.232166 
.231876 
.231586 
.231297 
.231008 
.230719 
.230429 
.230140 

0.229S52 
.229563 
.229274 
.228985 
.223697 
.228403 
.228120 
.227832 
.2275-13 
.227255 

0.226967 
.22G679 
.226392 
.226 1  ((4 
.22.5316 
.225.529 
.22.5241 
.224954 
.224667 
.224379 

0.224092 
.223305 
.223513 
.223232 
.222945 
.222658 
.222372 
.222035 
.221799 
.221512 
.221226 


Tang. 


M. 

60 
59 
58 
57 
56 
55 
54 
53 
62 
51 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
IS 
17 
16 
15 
14 
13 
12 
•1 

10 
9 
8 
7 
6 
5 
4 
3 
2 
I 
_0 

M. 


laoo 


COSINES,    TANGENTS,    AND    COTANGENTS. 


20e 

148= 


Sine. 


D.  1". 


0 
1 
2 
3 

4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 

ir> 

1(3 
17 
18 
19 

2(3 
21 
22 
23 
24 
25 
25 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 

41 

42 

43 

44 

45 

46 

47 

43 

49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


9  711839 
.712050 
.712260 
.712469 
.712679 
,712339 
.713093 
.713303 
.713517 
.713726 

9.713935 
.714144 
.714352 
.714561 
.714769 
.714978 
.715186 
.715394 
.715602 
.715309 

9.716017 
.716224 
.716132 
.710639 
.716346 
.7170.33 
.717259 
.717466 
.717673 
,717879 

9.7 1 8035 
.71S291 
.713497 
.718703 
.718909 
.719114 
.719320 
.719525 
.719730 
.719935 

9.720140 
.720345 
.720549 
.720754 
.720953 
.721162 
.721366 
.721570 
.721774 
.721978 

9.722181 
.722335 
.722588 
.722791 
.722994 
.723197 
.723400 
.723603 
.723305 
.724007 
.724210 


Cosiiie.   D.  1" 


3.50 
3.50 
3.50 
3.50 
3.49 
3.49 
3.49 
3.49 
3.43 
3.43 

3.43 
3.43 
3.43 
3.47 
3.47 
3.47 
3.47 
3.46 
3.46 
3.46 

3.46 
3.46 
3.45 
3.45 
3.45 
3.45 
3.44 
3.44 
3.44 
3.44 

3.43 
3.43 
3.43 
3.43 
3.43 
3.42 
3.42 
3.42 
3  42 
3.41 

3.41 
3.41 
3.41 
3.41 
3.40 
3.40 
3.40 
3.40 
3.39 
3.39 

3.39 
3.39 
3.39 
3.38 
3.38 
3.33 
3.33 
3.37 
3.37 
3.37 


9.933006 
.932990 
.932914 
.932833 
.932762 
.932685 
.932609 
.932533 
.932457 
.932330 

9.932304 
.932228 
.932151 
.932075 
.931998 
.931921 
.931845 
.931763 
.931691 
.931614 

9.931537 
.931460 
.931333 
.931306 
.931229 
.931152 
.931075 
.930993 
.930921 
.930343 

9.930766 
.930638 
.930611 
.930533 
.930456 
.930378 
.930.300 
.930223 
.930145 
.930067 

9.929939 
.929911 
.929333 
.929755 
.929677 
.929599 
.929521 
.929442 
.929364 
.929266 

9.929207 
.929129 
.929050 
.923972 
.923393 
.923315 
.923736 
.923657 
.923573 
.923499 
.923420 


Cosine.   D,  1" 


1.27 
1.27 
1.27 
1.27 

1.27 
1.27 
1.27 
1.27 
1.27 
1.27 

1.27 
1.27 
1.23 

1.28 
1.28 
1.28 
1.23 
1.23 
1.23 
1.23 

1.23 
1.23 
1.23 
1.23 
1.29 
1.29 
1.29 
1.29 
1.29 
1.29 

1.29 
1.29 
1.29 
1.29 
1.29 
1.29 
1.30 
1.30 
1.30 
1.30 

1.30 
1.30 
1,30 
1.30 
1.30 
1.30 
1.30 
1.31 
1.31 
1.31 

1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.31 
1.32 


Sine 


Tang. 

9.778774 
.779060 
.779346 
.779632 
.779918 
.7«0203 
.780489 
.780775 
.731060 
.781346 

9.781631 
.781916 
.782201 
.782186 
.782771 
.783056 
.783341 
.783626 
.783910 
.734195 

9  784479 
784764 
785043 
.785332 
.785616 
785900 
.786184 
.786463 
.786752 
.787036 

9.737319 
.787603 

.787886 
.783170 
.783453 
.783736 
.789019 
.789302 
.789535 
.7893C8 

9.790151 
.790434 
.790716 
.790999 
.791281 
.791563 
.791846 
.792128 
.792410 
.792692 

9.792974 
.793256 
.793533 
.793819 
.794101 
.794333 
.794664 
.794946 
.795227 
,795508 
.795739 


D.  1", 


D.  1".. 

4.77 
4.77 
4.77 
4.76 
4.76 
4.76 
4.76 
4.76 
4.76 
4.76 

4.75 
4.75 
4.75 
4.75 
4.75 
4.75 
4.75 
4.74 
4.74 
4.74 

4.74 
4.74 
4.74 
4.74 
4.73 
4.73 
4.73 
4.73 
4.73 
4.73 

4.73 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.72 
4.71 
4.71 

4.71 
4.71 
4.71 
4.71 
4.71 
4.70 
4.70 
4.70 
4.70 
4.70 

4.70 
4.70 
4.70 
4.69 
4.69 
4.69 
4.69 
4.69 
4.69 
4.69 


Cotang 


Cotang. 


0.221226 
.220940 
.220654 
.220363 
.220082 
.219797 
.219511 
.219225 
.218940 
.218654 

0.218369 
218034 
.217799 
.217514 
.217229 
.216944 
.216659 
.216374 
.216090 
.215805 

0.215521 
.215236 
.21 49-52 
.214663 
.214334 
.214100 
.21.3816 
.213532 
.213243 
.212964 

0.212681 
.212397 
.212114 
.211830 
.211547 
211264 
.210981 
.210698 
.210415 
.210132 

0.209849 
.209566 
.209284 
.209001 
.208719 
.208437 
.208154 
.207872 
.207590 
.207308 

0.207026 
.206744 
.206462 
.206181 
.205399 
.205617 
.20.3336 
.205054 
.204773 
.204492 
.204211 


M. 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


D.  1", 


Tang, 


M 


i»l^ 


5.>i^ 


206 

33° 


TABLE    Xlil.       LOGARITHMIC    SINES, 


M. 


0 
1 
2 
3 
4 
5 
6 
7 


10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

M. 


l«20 


Sine. 


9.724210 
.724412 
.724614 
.724S16 
.725017 
.725219 
.725420 
.725622 
.725523 
.726024 

9.726225 
.726426 
.726626 
.726327 
.727027 
.727228 
.727423 
.727623 
.727823 
.728027 

9.72S227 
.728427 
.725626 
.728325 
.729024 

.,_J29223 
.729422 
.729621 
.729320 
,730018 

9.730217 
.730415 
.730613 
.730811 
.731009 
.731206 
.731401 
.731602 
.731799 
.731996 

9.732193 
.732390 
.732537 
.732784 
.732930 
.733177 
.733373 
.733569 
.733765 
.733961 

9.7.34157 
.734353 
.734549 
.734744 
.734939 
.735135 
.735330 
.735525 
.735719 
.735914 
.736109 

Cosine. 


D.  1". 


3.37 
3.37 
3.36 
3.36 
3.36 
3.36 
3.36 
3.35 
3.35 
3.35 

3.35 
3.34 
3.31 
3.34 
3.34 
3.3i 
3.33 
3.33 
3.33 
3.33 

3.33 
3.32 
3.32 
3.32 
3.32 
3.31 
3.31 
3.31 
3.31 
3.31 

3.30 
3.30 
3.30 
3.30 
3.30 
3.29 
3.29 
3.29 
3.29 
3.28 

3.28 
3.28 
3.28 
3.28 
3.27 
3.27 
3.27 
3.27 
3.27 
3.26 

3.26 
3.26 
3.26 
3.26 
3.25 
3.25 
3.25 
3.25 
3.25 
3.24 

D.  1". 


Cosine. 


9.928420 

.923342 
.928263 
.923153 
.923104 
.925025 
.927946 
.927567 
.927787 
.927708 

9.927629 
.927549 
.927470 
.927390 
.927310 
.927231 
.927151 
.927071 
.926991 
.926911 

9.926331 
.926751 
.926671 
.926591 
.926511 
.926431 
.926351 
.926270 
.926190 
.926110 

9.926029 
.925949 
.925563 
.925733 
.925707 
.925626 
.925545 
.925465 
.925334 
.925303 

9.925222 
.925141 
.925060 
.924979 
.924397 
.924816 
.924735 
.924654 
.924572 
.924491 

9.924409 
.924328 
.924246 
.924164 
.924083 
.924001 
.923919 
.923837 
.923755 
.923673 
.923-591 

Sine. 


D.  1". 


1.32 
1.32 
1.32 
1.32 
1.32 
1.32 
L32 
1.32 
1.32 
1.32 

L32 
1.33 
1.33 
1.33 
1.33 
.33 
1.33 
1.33 
1.33 
1.33 

1.33 
1.33 
1.33 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 
1.34 

1.34 
1.34 
1.34 
1.34 
1.35 
1.35 
1.35 
1.35 
1.35 
1.35 

1.35 
1.35 
1.35 
1.35 
1.35 
1.35 
1.36 
1.36 
1.36 
1.36 

1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.36 
1.37 
1.37 
1.37 

D.  1". 


Tang. 


9.7957S9 
.796070 
.796351 
.796632 
.796913 
.797194 
.797474 
.797755 
.798036 
.798316 

9.798596 

.798877 
.799157 
.799437 
.799717 
.799997 
.800277 
.800557 
.800336 
.801116 

9.801396 
.801675 
.801955 
.802234 
.602513 
.802792 
.803072 
.803351 
.803630 
.803909 

9.804187 
.804466 
,804745 
.805023 
.805302 
.805580 
.805859 
.806137 
.806415 
.806693 

9.806971 
.807249 
.807527 
.807805 
.803033 
.803361 
.805633 
.803916 
.809193 
.809471 

9.809748 
.810025 
.810302 
.810580 
.810857 
.811134 
.811410 
.811687 
.811964 
.812241 
.812517 

Cotang 


D.  1", 


4.63 
4.68 
4.68 
4.68 
4.68 
4.63 
4.68 
4.68 
4.67 
4.67 

4.67 
4.67 
4.67 
4.67 
4.67 
4.66 
4.66 
4.66 
4.66 
4.66 

4.66 
4.66 
4.66 
4.65 
4.65 
4.65 
4.65 
4.65 
4.65 
465 

4.65 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4.64 
4.63 

4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.63 
4.62 
4.62 
4.62 

4.62 
4.62 
4.62 
4.62 
4.62 
4.61 
4.61 
4.61 
4.61 
4.61 

D.l" 


Cotang. 

M. 

0.204211 

60 

.203930 

59 

.203649 

58 

.203363 

57 

.203037 

56 

.202306 

55 

.202526 

54 

.202245 

53 

.201964 

52 

,201634 

51 

0.201404 

50 

.201123 

49 

.200843 

48 

.200563 

47 

.200283 

46 

.200003 

45 

.199723 

44 

.199413 

43 

.199164 

42 

.198884 

41 

0.195604 

40 

.19532.5 

39 

.195045 

38 

.197766 

37 

.197487 

36 

.197208 

35 

,196923 

34 

,196649 

33 

.196370 

32 

,196091 

31 

0.195813 

30 

.195534 

29 

,195255 

28 

,194977 

27 

,194698 

26 

.194420 

25 

,194141 

24 

.193363 

23 

.193555 

22 

.193307 

21 

0.193029 

20 

.192751 

19 

.192473 

IS 

.192195 

17 

.191917 

16 

.1916.39 

15 

.191362 

14 

.191084 

13 

.190807 

12 

.190529 

11 

0.190252 

10 

.189975 

9 

.189698 

8 

.189420 

7 

.189143 

6 

.188866 

5 

.188590 

4 

.188313 

3 

.188036 

2 

,187759 

1 

.187483 

0 

Tang 

M. 

COSINES,    TANGENTS,    AND    COTANGENTS. 


207 

1*1:0 


M 

0 

1 

2 
3 

4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine. 


D.  1". 


9.736109 
.736303 
.736493 
.736692 
.736SS6 
.737030 
.737274 
.737467 
.737661 
.737855 

9.733043 
.733241 
.733434 
.733627 
.733320 
.739013 
.739206 
.739393 
.739590 
.739783 

9.739975 
.740167 
.740359 
,740550 
.740742 
.740934 
.741125 
.741316 
.741508 
.741699 

9.741389 
.742080 
.742271 
.742462 
.742632 
.742S42 
.743033 
.743233 
.743413 
.743602 

9.743792 
.743932 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 
.745306 
.745594 

9.745633 
.745371 
.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747374 
.747562 


M. 


Cosine. 


Cosine. 


3.24 
3.24 
3.24 
3.23 
3.23 
3.23 
3.23 
3.23 
3.22 
3.22 

3.22 
3.22 
3.22 
3.21 
3.21 
3.21 
3.21 
3.21 
3.20 
3.20 

3.20 
3.20 
3.20 
3.19 
3.19 
3.19 
3.19 
3.19 
3.18 
3.18 

3.18 
3.18 
3.18 
3.17 
3.17 
3.17 
3.17 
3.17 
3.16 
3.16 

3.16 
3.16 
3.16 
3.15 
3.15 
3.15 
3.15 
3.15 
3.14 
3.14 

3.14 
3.14 
3.14 
3.13 
3.13 
3.13 
3.13 
3.13 
3.12 
3.12 


D.  1". 


9.923591 
.923509 
.923427 
.923345 
.923263 
.923181 
.923093 
.923016 
.922933 
.922351 

9,922768 
.922636 
.922603 
.922520 
.922433 
.9223.55 
.922272 
.9221S9 
.922106 
.922023 

9.921940 
.9218.57 
.921774 
.921691 
.921607 
.921524 
.921441 
.921357 
.921274 
.921190 

9.921107 
.921023 
.920939 
.920856 
.920772 
.9206S3 
.920604 
.920520 
.920436 
.920352 

9.920268 
.920184 
,920099 
.920015 
.919931 
.919346 
.919762 
.919677 
.919593 
.919503 

9.919424 
.919339 
.919254 
.919169 
.919035 
.919000 
.918915 
.918830 
.918745 
.918659 
.918574 


Tang. 


D.  1 '. 


D.  1". 


1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 
1.38 

1.33 
1.33 
1.33 
1.38 
1.3S 
1.38 
1.38 
1.38 
1.33 
1.33 

1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 
1.39 

1.39 
1.39 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 
1.40 

1.40 
1.40 
1.40 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 
1.41 

1.41 
1.41 
1.41 
1.41 
1.42 
1.42 
1.42 
1.42 
1.42 
1.42 


Sine 


9.312517 
.812794 
,813070 
.813347 
.813023 
.813399 
.814176 
.814452 
.814728 
.815004 

9.815230 
.815555 
.815-31 
.816107 
.816382 
.816653 
.816933 
.817209 
.8174.34 
.817759 

9.818035 
.818310 
.818585 
.818360 
.819135 
.819410 
.819634 
.8199.59 
.820234 
.820503 

9.820783 
.8210.57 
.821332 
.821606 
.821880 
.822154 
.822429 
.822703 
.822977 
.823251 

9.823524 

.823793 
.824072 
.824345 
.824619 
.824893 
.825166 
.825439 
.82.5713 
.825936 

9.8262.59 
.826532 
.826305 
.827078 
.827351 
.827624 
.827897 
.828170 
.828442 
.823715 
.823987 


Cotang. 


4.61 
4.61 
4.61 
4.61 
4.60 
4.00 
4.60 
4.60 
'1. 60 
4.60 

4.60 
4.60 
4.59 
4.^9 
4.59 
4.59 
4.59 
4.59 
4.59 
4.59 

4.59 
4..53 
4.53 
4.. 53 
4.58 
4.58 
4.-53 
4.53 
4.58 
4.58 

4.57 
4.-57 
4.57 
4.57 
4.-57 
4.57 
4.57 
4.57 
4.57 
4.56 

4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
4.56 
4.. 55 
4.55 

4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.54 
4.54 
4.54 


0.187483 
.187206 
.106930 
.186653 
.186377 
.186101 
.1S5>24 
.185543 
.185272 
.184996 

0.184720 
.184415 
.134169 
.183893 
.183618 
.183342 
.183067 
.182791 
.182516 
.182241 

0.181965 
.181690 
.131415 
.181140 
.180665 
.180590 
.180316 
.180041 
.179766 
.179492 

0.179r.l7 
.178943 
.178668 
.173394 
.173120 
.177846 
.177571 
.177297 
.177023 
.176749 

O.'l  76476 
.176202 
.175928 
.17.5655 
.175381 
.175107 
.174834 
.174,561 
.174287 
.174014 

0.173741 
.173468 
.173195 
.172922 
.172649 
.172.376 
.172103 
.171830 
.171558 
.171235 
.171013 


D.  1".   Cotang.  I  D.  1".    Tang. 


M. 

60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
33 
37 
36 
35 
34 
33 
32 
31 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
U 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


M. 


56 


i>08 

340 


TABLE    XIII.       LOGARITHMIC    SINES, 


1450 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
l-J 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 

52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine. 


9.747562 
,747749 
.747936 
.743123 
.743310 
,743497 
.743633 
.743370 
.749056 
.749243 

9.749123 
.749615 
.749301 
.749937 
.750172 
.750353 
.750543 
.750729 
.750914 
.751099 

9.751234 
.751469 
.751654 
.751839 
.752023 
.752203 
.752.392 
.752576 
.7.52760 
.752944 

9.753123 
.753312 
.753495 
.753679 
.753362 
.754046 
.75422J 
.751412 
.754595 
.754778 

9.754960 
.755143 
.755326 
.755503 
.755690 
.755372 
.756054 
.756236 
.756413 
.756600 

9.756782 
.756963 
.757144 
.757326 
.757507 
.757633 
.757869 
.753050 
.758230 
.753411 
.753591 


M.   Cosine. 


D.  1". 

3.12 
3.12 
3.12 
3.11 
3.11 
3.11 
3.11 
3.11 
3.K) 
3.10 

3.10 
3.10 
3.10 
3.10 
3.09 
3.09 
3.09 
3.09 
3.09 
3.03 

3.03 
3.03 
3.03 
3.03 
3.07 
3.07 
3.07 
3.07 
3.07 
3.06 

3.06 
3.06 
3.06 
3.06 
3.05 
3.05 
3.05 
3.05 
3.05 
3.05 

.3.04 
3.04 
3.04 
3.04 
3.04 
3.03 
3.03 
3.03 
3.03 
3.03 

3.02 
3.02 
3.02 
.3.02 
3.02 
3.02 
3.01 
3.01 
3.01 
3.01 


Cosine. 


D.  1". 


9.913574 
.91-4^9 
.918404 
.913313 
.913233 
.913147 
.913062 
.917976 
.917391 
.917805 

9.917719 
.917634 
.917548 
.917462 
.917376 
.917290 
.917204 
.917118 
.917032 
.916946 

9.916359 
.916773 
.916637 
.916600 
.916514 
.916427 
.916.341 
.916254 
.916167 
.916031 

9.915994 
.915907 
.91.5320 
.915733 
.915646 
.915.5.59 
.91.5472 
.915335 
.915297 
.915210 

9.915123 
.915035 
.914948 
.914360 
.914773 
.914635 
.914593 
.914510 
.914422 
.914334 

9.914246 
.914153 
.914070 
.913932 
.91.3394 
.913366 
.913718 
9136-30 
.913.541 
.913453 
.913365 


D.  1". 


Sine. 


1.42 
1.42 
1.42 
1.42 
1.42 
)  43 
1.43 
1.43 
1.43 
1.43 

1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.43 
1.44 
1.44 
1.44 

1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.44 
1.45 
1.45 

1.45 

1.45 

1.45- 

1.45 

1.45 

1.45 

1.45 

1,45 

1.45 

1.46 

1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1.46 
1,46 

1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 
1.47 


Tang. 


D.  1". 


9.8239S7 
.829260 
.829532 
.829 >05 
.830077 
.830349 

.Sb'06-4!l 
.830393 
.831165 
.8314.37 

9.831709 
.831931 
.832253 
.832.-,25 
.832796 
.833063 
.833339 
.8.3361 1 
.833332 
.834154 

9.331425 
.83 J 696 
.8.34967 
.8.35233 
.835509 
.835780 
.836051 
.836322 
.836593 
.836364 

9.837134 

.837405 
.837675 
.837946 
.833216 
.833487 
.838757 
.839027 
.839297 
.839563 

9.839833 
.840103 
.840378 
.840643 
.840917 
.841187 
.8414.57 
.841727 
.841996 
.842266 

9.842535 

.842305 
.843074 
.843343 
.84.3612 
.843332 
.844151 
.844420 
.844639 
.8449.58 
.84.5227 


D.  1". 


4.54 
4.. 54 
4.54 
4.54 
4.54 
4.54 
4.53 
4.53 
4.53 
4.53 

4.53 
4.53 
4.. 53 
4.53 
4.53 
4.53 
4.52 
4.52 
4.52 
4.52 

4.52 
4.52 
4.52 
4.52 
4..52 
4.52 
4.51 
4.51 
4.51 
4.51 

4.51 
4.51 
4.51 
4.51 
4.51 
4.51 
4.50 
4.50 
4.50 
4.50 

4.50 
4.50 
4.50 
4.50 
4.50 
4.49 
4.49 
4.49 
4.49 
4.49 

4.49 
4.49 
4.49 
4.49 
4.49 
4.49 
4.48 
4.48 
4.48 
4.43 


Cotang. 

M 

0.171013 

60 

.170740 

59 

.170463 

58 

.170195 

57 

.169923 

56 

.169651 

55 

.169379 

54 

.169107 

53 

.163335 

52 

.163563 

51 

0.168291 

50 

.163019 

49 

.167747 

48 

.167475 

47 

.167204 

46 

.166932 

45 

.166661 

44 

.166339 

43 

.166113 

42 

.165846 

41 

0.165575 

40 

.165301 

39 

Cotang.  :   D.  1". 


.165033 
.164762 
.164491 
.164220 
.16-3949 
.163678 
.16.3407 
.163136 

0.162S66 
,162-595 
.162-325 
.162054 
,161784 
,161513 
.161243 
.160973 
.160703 
.160432 

0.160162 
.1-59392 
.159622 
.159352 
.1.59083 
.153313 
,158543 
,1.58273 
.153004 
.1577'^ 

0.157465 
.1-57195 
.1-56926 
.156657 
.156-333 
.156118 
.1-5.5349 
.155530 
,15.53!! 

.154773 


Tang.     I  M. 


124c 


553 


COSINEb,    TANGENT&,    AND    COTANGENTS. 


209 

14:43 


M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


Sine. 

9.758591 
.758772 
.758952 
.759132 
.7o9:n2 
.759492 
.759672 
.759852 
.760:131 
.760211 

9.760390 
.760569 
.760748 
.760927 
.761106 
.761235 
.761464 
.761642 
.761821 
.761999 

9.7C2177 
.7623.")6 
.762534 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 

9.763954 
.764131 
.764308 
,764435 
.764662 
.7648.33 
.765015 
.765191 
.765.367 
.765544 

9.765720 
.765896 
.766072 
.766247 
.766423 
.766593 
.766774 
.766949 
.767124 
.767300 


D.l" 


50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60  1 

M.  I 


9.767475 
.767649 
.767824 
.767999 
.763173 
.768348 
.768522 
.763697 
.763371 
.769045 
.769219 


Cosine. 


3.01 
3.00 
3.00 
3.00 
3.00 
3.01) 
2.99 
2.99 
2.99 
2.99 

2.99 
2.99 
2.98 
2.98 
2.98 
2.93 
2.98 
2.97 
2.97 
2.97 

2.97 
2.97 
2.97 
2.96 
2.96 
2.96 
2.96 
2.96 
2.95 
2.95 

2.95 
2.95 
2.95 
2.95 
2.94 
2.94 
2.94 
2.94 
2.94 
2.93 

2.93 
2.93 
2.93 
2.93 
2.93 
2.92 
2.92 
2.92 
2.92 
2.92 

2.91 

2.91 

2.91 

2.91 

2.91 

2.91 

2.90 

2.90 

2.90 

2.90 


D.  1". 


9.913365 
.913276 
.913Ib7 
.9130'J9 
.913010 
.912922 
.912833 
.9127-14 
.912655 
.912566 

9.912477 
.912388 
.912299 
.912210 
.912121 
.912031 
.911942 
.911853 
.911763 
.911674 

9.911584 
.911495 
.911405 
.911315 
.911226 
.911136 
.911046 
.910956 
.910866 
.910776 

9.910636 
.91U596 
.910506 
.910415 
.910325 
.910235 
.910144 
.910054 
.909963 
.909873 

9.9097S2 
.909691 
.909601 
.909510 
.909419 
.909328 
.909237 
.909146 
.909055 
.903964 

9.908873 
.903781 
.908690 
.903599 
.903507 
.908416 
.903324 
.903233 
.903141 
.903049 
.907958 


Cosine.   D.  1". 


Tang. 


D.  1". 


1.47 
1.48 
1.43 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 

1.48 
1.48 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 
1.49 

1.49 
1.49 
1.49 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 

I.. 50 
1.50 
1.50 
1.51 
1.51 
1.51 
1.51 
1.51 
1.51 
1.51 

1.51 
1.51 
1.51 
1.51 
1.52 
1.52 
1.52 
1.52 
1.52 
1.52 

1.52 
1.52 
1.52 
1.52 
1.52 
1.53 
1.53 
1.53 
1..53 
1.53 


Sine. 


9.845227 
.845496 
.845764 
.846033 
.846302 
.846370 
.846839 
.847108 
.847376 
.847644 

9.847913 

.843181 
.843449 
.848717 
.843986 
.849254 
.849522 
.849790 
.850057 
.350325 

9.850593 
.850861 
.851129 
.851.396 
.851664 
.851931 
.852199 
.852466 
.852733 
.853001 

9.853268 
.853535 
.853302 
,854C69 
.854336 
.854603 
.854870 
.855137 
.8.55404 
.855671 

9.855933 
.856204 
.8.56471 
.856737 
.857004 
.857270 
.857537 
.857803 
.858069 
.858336 

9.85S602 
.858868 
.859134 
.859400 
.859666 
.859932 
.860198 
.860464 
.860730 
.860995 
.861261 


Cotang. 


4.48 
4.48 
4.48 
4.43 
4.48 
4.48 
4.48 
4.47 
4.47 
4.47 

4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.47 
4.46 
4.46 
4.46 

4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.46 
4.45 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.44 

4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 
4.44 

4.44 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 
4.43 


0.154773 
.154504 
.154236 
.153967 
.153698 
153430 
.153161 
.152892 
.152624 
.152356 

0.152087 
.151819 
.151551 
.151283 
.151014 
.150746 
.150478 
.150210 
.149943 
.149675 

0.149407 
.149139 
.148871 
.148604 
.148336 
.148C69 
.147801 
.147534 
.147267 
.146999 

0.146732 
.146465 
.146198 
.145931 
.145664 
.145397 
.145130 
.144363 
.144596 
.144329 

0.144062 
.143796 
.143.529 
.143263 
.142996 
.142730 
.142463 
.142197 
.141931 
.141664 

0.141398 
.141132 
.140866 
.140600 
.140334 
.140063 
.139802 
.1395.36 
.139270 
.139005 
.138739 


D.  1".  I  Cotang.   D.  1".  I   Tang. 


M. 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 


210 

B60 


TABLE    XIII. 


LOGARITHMIC    SINES, 


14:3 


M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


9.769219 
.769393 
.769566 
.769740 
.769913 
.770037 
.770260 
.770433 
.770506 
.770779 

9.770952 
.771125 
.771293 
.771470 
.771843 
.771815 
.771937 
.772159 
.772331 
.772503 

9.772675 
.772847 
.773013 
.773190 
.773361 
.773533 
.773704 
.773575 
.774046 
,774217 

9.774338 

.77455S 
,774729 
,774399 
.775070 
775240 
.775410 
.775530 
.775750 
.775920 

9.776090 
.776259 
.776429 
.776593 
,776763 
.776937 
,777106 
,777275 
,777444 
.777613 

9.777731 
.777950 
.773119 

.778237 
.773455 
.773624 
.773792 
.773960 
.779123 
.779293 
.779463 

Cosine. 


D.  1". 


2.00 
2.90 
2.39 
2.39 
2.39 
2.89 
2.89 
2,33 
2.33 
2.83 

2.88 
2.33 
2.33 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.S6 

2.36 
2.36 
2.36 
2.86 
2.35 
2.35 
2.85 
2.85 
2.35 
2.35 

2.34 
2.84 
2.34 
2.34 
2.34 
2.84 
2.S3 
2.33 
2.33 
2.83 

2.83 
2.83 
2.32 

2.32 
2.82 
2.82 
2.82 
2.82 
2.81 
2.81 

2.81 
2.81 
2.81 
2.81 
2.  SO 
2.80 
2.80 
2.80 
2.S0 
2.79 

D.  1". 


Cosine. 


9.907953 
.907866 
.907774 
.907632 
.907590 
.907493 
.907406 
.907314 
.907222 
.907129 

9.907037 
.906945 
.906352 
.906760 
.906667 
.906575 
.906432 
.906339 
.906296 
.906204 

9.906111 
.906018 
.905925 
.90.5332 
.905739 
.905645 
.905552 
.905459 
.905366 
.905272 

9.905179 
.905035 
.904992 
.904893 
.904304 
.904711 
.904617 
.90^1523 
,904429 
.904335 

9.904241 
.904147 
,904053 
.903959 
.903364 
.903770 
.90.3676 
.903.581 
.90.3487 
.903392 

9.903293 
,903203 
.903103 
.903014 
.902919 
.902324 
,902729 
.9026.34 
.902.539 
,902444 
.902349 

Sine. 


D.  1". 


,53 
,53 
,53 
,53 
,53 
,53 
,54 
,54 
.54 
54 

,54 
,54 
.54 
,54 
54 
.54 
,55 
55 
,55 
,55 

,55 
,55 
,55 
,55 
,55 
,55 
,55 
,56 
,56 
,56 

,.56 
,56 
.56 
56 
56 
,56 
,56 
,57 
,57 
57 

,57 
,57 
,57 
57 
57 
,57 
,57 
.57 
,58 
,58 

.58 
,58 
,58 
,53 
,53 
,.53 
,58 
,53 
.59 
,59 


D.  1". 


Tang. 


9.861261 
.861.527 
.861792 
.862058 
,862323 
.862.589 
,862354 
,863119 
,863335 
.86.3650 

9.863915 
.864180 
,864445 
,864710 
.864975 
.865240 
,865.505 
.865770 
.866035 
.866300 

9.866564 
,866329 
.867094 
,8673.58 
.867623 
.867337 
.8681.52 
.863416 
.86S630 
.863945 

9.869209 
.869473 
.869737 
.870001 
.870265 
.870529 
,870793 
,871057 
,871321 
.871585 

9.871349 
.872112 
.872376 
,872640 
.872903 
.873167 
.873430 
.873694 
.873957 
.874220 

9.874434 
.874747 
.875010 
.875273 
.8755.37 
.875300 
.876063 
.876326 
.876589 
,376352 
■877114 

Cotang, 


D.  1". 


4.43 
4.43 
4.43 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 
4.42 

4.42 
4.42 
4.42 
4.42 
4.42 
4.41 
4.41 
4.41 
4.41 
4.41 

4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.41 
4.40 
4.40 

4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 

.  4.40 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 
4.39 

4.39 
4.39 
4.39 
4.39 
4.33 
4.33 
4.33 
4.33 
4.38 
4.33 

D.  1". 


Cotang. 


0.133739 
,133173 
133203 
,137942 
,137677 
,137411 
137146 
,136881 
,136615 
,136350 


0 


136085 
135820 
135555 
135290 
135025 
134760 
134495 
1342.30 
13.3965 
133700 

133436 
133171 
132906 
132642 
132.377 
132113 
131843 
131534 
131320 
131055 

130791 
130527 
130233 
129999 
129735 
129471 
129207 
123943 
123679 
123415 

128151 
127883 
127624 
127360 
127097 
126833 
126570 
126306 
126043 
125780 

125516 
125253 
124990 
124727 
124463 
124200 
123937 
123674 
12311 1 
123143 
122386 

Tang. 


ISd^ 


63 


COSINES,    TANGENTS,    AND    COTANGENTS. 


211 

1433 


M. 


Sine. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 

12 
13 

14 
15 
16 
[7 
IS 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
3i 
35 
36 
37 
35 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
o7 
5S 
59 
60 


D.  1". 


9.779463 
.779631 

.779798 
.779966 
.780 133 
.780300 
.78(H67 
,780634 
.780801 
.780968 

9.781134 
,781301 
.781463 
.781634 
.781800 
.731966 
.782132 
.782293 
.782464 
.782630 

9.782796 
.782961 
.783127 

,783292 
.783453 
.783623 
,783783 
,783953 
.784118 
,784282 

9.784447 
.784612 
.784776 
.784941 
.785105 
.785269 
.785433 
.785597 
.785761 
.785925 

9.736039 
.7862.52 
.786416 
.786579 
.786742 
.736906 
.737069 
.787232 
.787395 
.787557 

9.787720 
.737833 
.738045 
.733208 
.733370 
.783532 
.783694 
.783856 
.789018 
.789180 
.739^12 


Cosine. 


2.79 
2.79 
2.79 
2.79 
2.79 
2.78 
2.78 
2.78 
2.73 
2.73 

2.78 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.76 
2.76 
2.76 

2.76 
2.76 
2.76 
2.75 
2.75 
2.75 
2.75 
2.75 
2.75 
2.74 

2.74 
2.74 
2.74 
2.74 
2.74 
2.73 
2.73 
2.73 
2.73 
2.73 

2.73 
2.73 
2.72 
2.72 
2.72 
2.72 
2.72 
2.72 
2.71 
2.71 

2.71 

2.71 

2.71 

2.71 

2.70 

2.70 

2.70 

2.70 

2.70 

2.70 


D.  1". 


9.902349 
.902253 
.902158 
.902063 
.901967 
.901372 
.901776 
.901681 
.901585 
.901490 

9.901394 
.901293 
.901202 
.901106 
.901010 
.900914 
.900818 
.900722 
.900626 
.900529 

9.900433 
.900337 
.900240 
.900144 
.900047 
.899951 
.899354 
.899757 
.899660 
.899564 

9.899467 
.899370 
.899273 
.899176 
.899073 
.893981 
.893834 
.893787 
.898689 
.893592 

9.898494 
.898397 
.893299 
.898202 
.893104 
.893006 
.897908 
.897810 
.897712 
.897614 

9.897516 
.897418 
.897320 
.897222 
.897123 
.897025 
.896926 
.896828 
.896729 
.896631 
.896532 


Tang. 


1.59 
I. .59 
1.59 
1.59 
1.59 
1.59 
1..59 
1.59 
1.59 
1.60 

1.60 
1.60 
1.60 
1.60 
l.GO 
1.60 
1.60 
1.60 
1.60 
1.61 

1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.61 
1.62 

1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 
1.62 

1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 

1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 
1.64 


M.       Cosine.    I   D.  1".  I      Sine.        D.  1".      Cotang. 


D.  1". 


9.877114 
.877377 
.877640 
.8779ft3 
.873165 
.878423 
.878691 
.878953 
.879216 
.879478 

9.879741 

.880003 
.880265 

.880528 
.880790 
.8.^5  1052 
881314 
.881577 
.881839 
.882101 

9.8S23e3 
.882625 
.832887 
.883143 
.883410 
.883672 
.883934 
.834196 
.884457 
.884719 

9.884930 
.835242 
.885504 
.835765 
.886026 
.886283 
.886.549 
.886811 
.887072 
.887333 

9.887594 
.837855 
.883116 
.883378 
.838639 
.888900 
.889161 
.889421 
.839682 
.889943 

9.890204 
.890465 
.890725 
.8909-^6 
891247 
.891507 
.891763 
.892023 
.892239 
.892549 
.892310 


Cotang. 


4.38 
4.38 
4.38 
4.38 
4.38 
4.38 
4.38 
4.33 
4.. 37 
4.37 

4.37 
4.37 
4.37 
4.37 
4.-37 
4.37 
4.37 
4.37 
4.37 
4.37 

4.37 
4.37 
4.36 
4. -36 
4.36 
4.-36 
4.36 
4.36 
4.36 
4.30 

4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.36 
4.35 
4.35 
4.35 

4.35 
4.35 
4.-35 
4.35 
4.. 35 
4.35 
4.35 
4.35 
4.35 
4.35 

4.35 
4.35 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 


M. 


0.122886 
.122623 
.122360 
.122097 
.121835 
.121572 
.121309 
.121047 
.120784 
.120522 

0.1202.59 
.119997 
.119735 
.119472 
.119210 
.118943 
.118686 
.118423 
.118161 
.117899 

0.1176.37 
.117375 
.117113 
.116852 
.116590 
.116328 
.116066 
.115804 
.115543 
.115281 

0.115020 
.114758 
.114496 
.114235 
.113974 
.11.3712 
.113451 
.113189 
.112928 
.  1 126G7 

0.112406 
.112115 
.111 S84 
.111(;22 
.111361 

.imon 

.I1(K:!9 
.11(1579 
.11(1318 
.1  10057 

0.  l(l'.)796 
.l(i',).'):'.5 
.l!n)275 
.109014 
.1087 .53 
.108493 
.108232 
.107972 
.107711 
.107451 
.107190 


D.  1". 


60 
59 
58 
57 
56 
55 


52 
51 

50 
•49 
48 
47 
46 
^5 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
IS 
17 
10 
15 
14 
13 
12 
II 

1ft 
9 

8 


Tang.   M 


laT' 


5a<- 


212 

38° 


TABLE    XIIT. 


LOGARITHMIC    SINES, 


14:1C 


M. 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
II 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
31 
35 
36 
37 
38 
39 

40 
41 

42 
43 
44 
45 

46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


Sine. 


D.  1". 


9.789312 
.789504 
.789665 
.789S27 
.789983 
.790149 
.790310 
.790471 
.790632 
.790793 

9.790954 
.791115 
.791275 
.791436 
.791596 
.791757 
.791917 
.792077 
.792237 
.792397 

9.7925.57 
.792716 
.792376 
.793035 
.793195 
.793354 
.793514 
.793673 
.793832 
.793991 

9.794150 
.794303 
.794467 
.794626 
.794784 
.794942 
.795101 
.795259 
.795417 
.795575 

9.795733 
.795891 
.796049 
.796206 
.796.364 
.790521 
.796679 
.796836 
.796993 
.797150 

9.797.307 
.797464 
.797621 
.797777 
.797934 
.793091 
.798247 
.793403 
.798560 
,798716 
.793872 


M.      Cosine. 


2.69 
2.69 
2.69 
2.69 
2.69 
2.69 
2.6S 
2.63 
2.68 
2.63 

2.63 
2.63 
2.67 
2.67 
2.67 
2.67 
2.67 
2.67 
2.67 
2.66 

2.66 
2.66 
2.66 
2.66 
2.66 
2.65 
2.65 
2.65 
2.65 
2.65 

2.65 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.64 
2.63 
2.63 

2.63 
2.63 
2.63 
2.63 
2.62 
2.62 
2.62 
2.62 
2.62 
2.61 

2.61 
2.61 
2.61 
2.61 
2.61 
2.61 
2.61 
2.60 
2.60 
2.60 


Cosine. 


D.l 


9.896.5-32 
.896433 
.896335 
.896236 
.896137 
.896038 
.895939 
.895840 
.895741 
.895641 

9.895542 
.89.5443 
.895343 
.895244 
.895145 
.895045 
.894945 
.894846 
.894746 
.894646 

9.894.546 
.894446 
.894346 
.894246 
.894146 
.894046 
.893946 
.893846 
.893745 
.893645 

9.893544 
.893444 
.893343 
.893243 
.893142 
.893041 
.892940 
.8923.39 
.892739 
.892633 

9.892536 
.892435 
.892334 
.892233 
.892132 
.892030 
.891929 
.891827 
.891726 
.891624 

9.891523 
.891421 
.891319 
.891217 
.891115 
.891013 
.890911 
.890809 
.890707 
.890605 
.890503 


D.  1". 


Sine. 


1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 
1.65 

1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 
1.66 

1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 
1.67 

1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 
1.63 

1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 
1.69 

1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 
1.70 


Tang. 


9.892S10 
.893070 
.893331 
.893591 
.893851 
.894111 
.894372 
.894632 
.894892 
.895152 

9.895412 
.895672 
.895932 
.896192 
.8964.52 
.896712 
.896971 
.897231 
.897491 
.897751 

9.898010 
.898270 
.8985.30 
.898789 
.899049 
.899308 
.899563 
.899827 
.900087 
.900346 

9.900605 
.900864 
.901124 
.901383 
.901642 
.901901 
.902160 
.902420 
.902679 
.902933 

9.303197 
.903456 
.903714 
.903973 
.9042.32 
.904491 
.904750 
.905003 
.805267 
.905526 

9.905785 
.906043 
.906302 
.906560 
.906819 
.907077 
.907336 
.907594 
.907853 
.903111 
.908269 


D.  1". 


D.  1".   Cotang. 


4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.34 
4.33 
4.33 

4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.-33 
4.33 

4.33 
4.33 
4.33 
4.-33 
4.33 
4.32 
4.32 
4.32 
4.32 
4.-32 

4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 
4.32 

4.32 
4.32 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 

4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 
4.31 


Cotang. 

M. 

0.107190 

60 

.106930 

59 

.106669 

58 

.106409 

57 

.106149 

56 

.105889 

55 

.105628 

54 

.10.5368 

53 

.105108 

52 

.104848 

61 

0.104583 

50 

.104328 

49 

.104063 

48 

.103808 

47 

.103548 

46 

.103283 

45 

.103029 

44 

.102769 

43 

.102509 

42 

.102249 

n  I ni nnn 

41 

D.  1". 


.101730 
.101470 
.101211 
.100951 
.100692 
.100432 
.100173 
.099913 
.099654 

0.099395 
.099136 
.098876 
.098617 
.098358 
.098099 
.097840 
.097580 
.097321 
.097062 

0.096S03 
.096544 
.096286 
.096027 
.095768 
.095509 
.095250 
.094992 
.094733 
.094474 

0.094215 

.09.3957 
.093698 
.093440 
.093181 
.092923 
.092664 
.092406 
.092147 
.091889 
.091631 


Tang.   M 


10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


l»8o 


61 


COSINES,    TANGENTS,    AND    COTANGENTS. 


2Vc 

14:0= 


M. 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
53 
57 
5.3 
59 
60 


Sine. 


D.  1". 


9.793372 
.799023 
.799134 
.799339 
.79949.5 
.799651 
.799306 
.799962 
.800117 
.800272 

9.300427 
.800532 
.S0J737 
.800392 
.801047 
.801201 
.8013.56 
.801511 
.801665 
.801819 

9.801973 
.802123 

.802232 
,802436 
.802589 
.802743 
.802397 
.803050 
.803204 
.803357 

9.803511 

.803664 
.803817 
.803970 
.804123 
.804276 
.804423 
.804531 
.804734 
.804836 

9.805039 
.805191 
.80.5343 
.805495 
.80.5647 
.805799 
.80.5951 
.806103 
.8062.54 
.806406 

9.806557 
.806709 
.806360 
.80701 1 
.807163 
.8(37314 
.807465 
.807615 
.807766 
.807917 
.803067 


Cosine. 


2.60 
2.60 
2.6') 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 
2.59 

2.58 
2.53 
2.58 
2.58 
2.  .58 
2.58 
2.57 
2.57 
2.57 
2.57 

2.57 
2.57 
2.57 
2.. 56 
2.56 
2.56 
2.56 
2.56 
2.56 
2.55 

2.55 
2.55 
2.55 
2.  .55 
2.55 
2.. 55 
2.54 
2.54 
2.54 
2.54 

2.54 
2.54 
2.54 
2.53 
2.53 
2.  .53 
2.  .53 
2.. 53 
2.53 
2.52 

2.52 
2.52 
2.52 
2.  .52 
2.. 52 
2.52 
2.51 
2.51 
2.51 
2.51 


M. 


LS9^ 


Cosine. 


D.  1". 


9.890503 
.890400 
.890293 
.890195 
.890093 
.839990 
.889333 
.889785 
.889632 
.839579 

9.839477 
.839374 

.889271 
.839163 
.889064 
.838961 
.833853 
.833755 
.888651 
.838543 

9.833444 
.833341 

.838237 
.833134 
.888030 
.887926 
.887822 
.837718 
.837614 
.837510 

9.837406 

.837302 
.837198 
.837093 
.836'989 
.836385 
.836780 
.836676 
.886571 
.886466 

9.836362 

.836257 
.836152 
.836047 
.835942 
.835337 
.835732 
.835627 
.835.522 
.835416 

9.83.5311 

.83.5205 
.835100 
.884994 
.834339 
.834783 
.834677 
.834572 
.834466 
.834360 
.884254 


D.  1" 


Tang. 


D.  1". 


1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 
1.71 

1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 
1.72 

1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.73 
1.74 

1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.74 
1.75 

1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.75 
1.76 

1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.76 
1.77 
1.77 


Sine. 


9.903369 
.903628 
.90S336 
.909144 
.909402 
.909660 
.909918 
.910177 
.910435 
.910693 

9.910951 
.911209 
.911467 
.911725 
.911932 
.912240 
.912493 
.912756 
.913014 
.913271 

9.913529 
.913787 
.914044 
.914302 
.914.560 
.914317 
.915075 
.91.5332 
.915.590 
.915347 

9.916104 
.916362 
.916619 
.916877 
.917134 
.917391 
.917648 
.917906 
.918163 
.918420 

9.918677 
.918934 
.919191 
.919448 
.919705 
.919962 
.920219 
.920476 
.920733 
.920990 

9.921247 
.921503 
.921760 
.922017 
.922274 
.922530 
.922787 
.923044 
.923300 
.923557 
.923314 


Cotang. 


4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 

4.30 
4.30 
4.30 
4.. 30 
4.30 
4.30 
4.30 
4.30 
4.30 
4.30 

4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 

4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 
4.29 

4.23 
4.23 
4.28 
4.28 
4.23 
4.23 
4.23 
4.28 
4.23 
4.28 

4.28 
4.23 
4.28 
4.28 
4.28 
4.23 
4.23 
4.28 
4.23 
4.23 


D.  1".  I  Coteng. 


0.091631 
.091372 
.091114 
.090356 
.090598 
.090340 
.090032 
.089323 
.039565 
.039307 

0.0S9049 
.038791 
.038533 
.083275 
.033018 
.037760 
.037502 
.037244 
.036936 
.086729 

0.036471 
.086213 
.0359.56 
.085693 
.085440 
.035183 
.084925 
.084663 
.084410 
.034153 

0.033396 
.033633 
.033331 
.033123 
.032366 
.082609 
.032352 
.032094 
.031337 
.081580 

0.031323 
.031066 
.080309 
.030552 
.030295 
.030033 
.079781 
.079524 
.079267 
.079010 

0.073753 
.073497 
.078240 
.077933 
.077726 
.077470 
.077213 
.076956 
.076700 
.076443 
.076136 


M. 

60 


D.  1". 


Tang. 


59 
53 
57 
56 
55 
.54 
53 
52 
51 


49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
23 
27 
26 
25 
24 
23 
22 
21 

20 
19 

18 
17 
16 
15 
14 
13 
12 
11 

10 
9 
8 
7 
6 
5 
4 
3 
2 
I 
_0_ 

M. 


tu^. 


214 

*0O 


TABLE    XIII.       LOGARITHMIC    SINES, 


139" 


M. 

0 
I 

2 
3 

4 
5 
6 

7 
8 
9 

10 
II 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 

44 
45 
46 
47 
,48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

M. 


Sine. 


9.808067 
.808218 
.803363 
.808519 
.808669 
.808819 
.808969 
.809119 
.809269 
.809419 

9.809569 
.809713 
.809363 
.810017 
.810167 
.810316 
.810465 
.810614 
.810763 
.810912 

9.311061 
.811210 
.8113.53 
.811507 
.811655 
.811804 
.811952 
.812100 
.812243 
,812396 

9.812544 
312692 
.812S40 
.812933 
.813135 
.8132S3 
.813430 
.813573 
.813725 
.813872 

9.814019 
.814166 
.814313 
.814460 
.814607 
.814753 
.814900 
.815046 
.815193 
.815339 

9.815185 
.815632 
.815773 
.815924 
.816069 
.816215 
.816361 
.816.507 
.816652 
.816793 
.816943 


D.  1". 


2.51 
2.51 
2.51 
2.50 
2.50 
2.50 
2.50 
2.  .50 
2.50 
2.50 

2.49 
2.49 
2.49 
2.49 
2.49 
2.49 
2.43 
2.43 
2.43 
2.43 

2.43 
2.43 
2.43 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 
2.47 

2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.46 
2.45 
2.45 
2.45 

2.45 
2.45 
2.45 
2.45 
2.44 
2.44 
2.44 
2.44 
2.44 
2.44 

2.44 
2.43 
2.43 
2.43 
2.43 
2.43 
2.43 
2.43 
2.42 
2.42 


Cosine 


D.  1". 


Cosine. 


9.8342-54 
.831143 
.881042 
.883936 
.883329 
.833723 
.833617 
.833510 
.883401 
.883297 

9.883191 

.883034 
.882977 
.882371 
.882764 
.882657 
.882550 
.882443 
.8823:36 
.882229 

9  882121 

.882014 
.831907 
.831799 
.831692 
.881534 
.881477 
.831369 
.831261 
.831153 

9.831046 

.880933 
.830330 
.830722 
.850613 
.830.505 
.830397 
.880239 
.830180 
.830072 

9.379963 

.879355 
.879746 
.879637 
.879529 
.879120 
.879311 
.879202 
.879093 
.873984 

9.878375 
.878766 
.878656 
.878547 
.87,3433 
.878323 
.873219 
.878109 
.877999 
.877890 
.877780 


Sine. 


D.  1". 


.77 
.77 
.77 
.77 
.77 
.77 
.77 
.77 
.73 
.73 

.73 
.78 
.78 
.73 
.78 
.78 
.73 
.79 
.79 
.79 

.79 
.79 
.79 
.79 
.79 
.79 
.79 
.80 
.80 
.80 

.80 
.80 
.80 
.80 
.80 
.80 
.31 
.31 
.81 
.81 

.81 

.81 
.81 
.81 

.81 
.81 

.82 
.82 
.82 
.82 

.82 
.82 
.82 
.82 
.82 
.83 
.83 
.83 
.83 
.83 


D.  1". 


Tang. 


9.92.3314 
.921070 
.924327 
.921583 
.924840 
.925096 
.925352 
.925609 
.92-5865 
.926122 

9.926373 
.9266.34 
.926890 
.927147 
.927403 
.927659 
.927915 
.928171 
.923127 
.92?634 

9.923940 
.929196 
.929452 
.929703 
.929964 
.930220 
.930475 
.930731 
.930937 
.931243 

9.931199 
.931755 
.932010 
.932266 
.932522 
.932773 
.933033 
.933239 
.93.3545 
.933800 

9.931056 
.9.31311 
.9.31567 
.9.31322 
.9.35078 
.935.3.33 
.935.539 
.935314 
.936100 
.9363.55 

9.936611 
.936366 
.937121 
.937377 
.937632 
.937837 
.933142 
.938393 
.9336.53 
.933903 
.939163 


D.  1". 


4.28 
4.23 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 

4.27 

4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.27 

4.27 
4.27 
4.27 
4.27 
4.27 
4.27 
4.26 
4.26 
4.26 
4.26 

4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 

4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 
4.26 

426 
4.26 
4.26 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 


Cotang.  i  D.  1". 


Cotang. 


0.076186 
.075930 
.075673 
.075417 
.075160 
.074901 
.074613 
.071391 
.074135 
.073878 

0.073622 
.073366 
.073110 
.072853 
.072597 
.072.341 
.072135 
.071329 
.071573 
.071316 

0.071060 
.070301 
.070.548 
.070292 
.070036 
.069730 
.069525 
.069269 
.069013 
.063757 

0.068501 
.063215 
.067990 
.067731 
.067478 
.067222 
.066967 
.066711 
.06&155 
.066200 

0.06.5944 
.065639 
.0654.33 
.065178 
.061922 
.061667 
.064411 
.0641.56 
.063900 
.063645 

0.063.389 
.063134 
.062379 
.062623 
.062363 
.062113 
.061358 
.061602 
.061347 
.061092 
.060837 


Tang.  I  M. 


I9f%0 


49^ 


COSlNEll,    TANGENTS,    AND    COTANGENTS. 


410 


215 

1383 


M. 

0 
1 
•2 
3 
4 
5 
6 
7 
S 
9 

10 
U 
12 
13 
14 
15 
16 
17 
IS 
19 

20 

21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 


54 
55 
56 
57 
53 
59 
60 

M. 


Sine. 


9.816943 

.817038 
.817233 
.817379 
.817524 
.817663 
.817813 
.817953 
.818103 
.818247 

9.818392 
.818536 
.818631 
.81S325 
.818969 
.319113 
.819257 
.319401' 
.819545 
.3196S9 

9.819832 
.819976 
.820120 
.820263 

.820406 
.820550 
.820693 
.S20S36 
.820979 
.821122 

9.321265 
.821407 
.821550 
.321693 
.321335 
.321977 
.822120 
.822262 
.822404 
.822546 

9.822633 
.322830 
.822972 
.323114 
.823255 
.82.3397 
.823539 
.82.3680 
.823821 
.823963 

9.824104 
.824245 
.824386 
.324527 
.824668 
.824303 
.824949 
.8-25090 
.82.5230 
.825371 
.825511 

Cosine. 


D.  1". 


2.42 
2.42 
2.42 
2.42 
2.42 
2.41 
2.41 
2.41 
2.41 
2.41 

2.41 
2.41 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.40 
2.39 

2.39 
2.39 
2.39 
2.39 
2.39 
2.39 
2.33 
2.-33 
2.38 
2.38 

2..33 
2.38 
2.33 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 
2.37 

2.37 
2.. 36 
2.36 
2.. 36 
2.. 36 
2.. 36 
2..36 
2.36 
2.35 
2.35 

2.35 
2.35 
2.35 
2.35 
2.35 
2.34 
2.34 
2.34 
2.34 
2.34 

D.  1". 


Cosine. 


9.877780 
.877670 
.877560 
.877450 
.877340 
.877230 
.877120 
.877010 
.876399 
.876739 

9.376678 
.876568 
.876457 
.876347 
.876236 
.876125 
.876014 
.875904 
.875793 
.875682 

9.875571 

.875459 
.875.348 
.8752:37 
.875126 
.875014 
.874903 
.874791 
.874680 
.874568 

9.874456 
.874344 
.874232 
.874121 
.874009 
.873396 
.873734 
.873672 
.873560 
.873443 

9.873335 
.873223 
.873110 
.872993 

.872385 
.872772 
.872659 
.872.547 
.3724.34 
.872321 

9.872203 
.872095 
.871981 
.871863 
.871755 
.871641 
.871528 
.871414 
.871301 
.871137 
.871073 

Sine. 


D.  1". 


1.83 
1.83 
1.83 
1.83 
1.84 
1.34 
1.84 
1.84 
1.84 
1.34 

1.84 
1.84 
1.84 
1.84 
1.85 
1.85 
1.85 
1.85 
1.85 
1.85 

1.85 
1.85 
1.85 
1.86 
1.86 
1.86 
1.86 
1.86 
1 .86 
1.86 

1.86 
1.86 

1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 
1.87 

1.87 
1.88 
1.83 
1.88 
1.83 
1.83 
1.88 
1.88 
1.88 
1.88 

1.89 
1.89 
1.89 
1.89 
1.89 
i.89 
1.89 
1.89 
1.89 
1.90 

D.  1". 


Tang. 


9.939163 
.939418 
.939673 
.939923 
.940183 
.940439 
.940694 
.940949 
.941204 
.941459 

9.941713 
.941968 
.942223 
.942478 
.942733 
.942988 
.943243 
.943493 
.943752 
.944007 

9.944262 
.944517 
.944771 
.94.5026 
.945281 
.945535 
.945790 
.946045 
.946299 
.946554 

9.946S08 
.947063 
.947318 
.947572 
.947827 
.948031 
.948335 
.948590 
.948344 
.949099 

9.949353 
.949603 
.949862 
.950116 
.950371 
.950625 
.950879 
.951133 
.951383 
.951642 

9.951896 
.952150 
.952405 
.952659 
.952913 
.953167 
.953421 
.953675 
.953929 
.954183 
.954437 

Cotang. 


D.  1'. 


4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 
4.25 

4.25 
4.25 
4.24 
4.24 
4.24 
4.24 
4.21 
4.24 
4.24 
4.24 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 

4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.24 
4.23 
4.23 
4.23 

D.  1". 


Cotang. 


0.060837 
.060582 
.060327 
.060072 
.059817 
.059561 
.059306 
.059051 
.058796 
.058541 

0.053287 
.053032 
.057777 
.057522 
.057267 
.057012 
.056757 
.056.502 
.056248 
.055993 

0.055733 
.0.55483 
.055229 
.054974 
.054719 
.054465 
.0.54210 
.053955 
.05.3701 
.053446 

0.053192 
.052937 
.052682 
.052128 
.052173 
.051919 
.051665 
.051410 
.051156 
.050901 

0.050647 
.050392 
.050138 
.049834 
.049629 
.049375 
.049121 
.043367 
.048612 
.048358 

0.043104 
.047850 
.047595 
.047341 
.047037 
.046333 
.046579 
.046325 
.046071 
.045817 
.045563 


M. 

60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
43 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
13 
17 
16 
15 
14 
13 
12 

ir 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


Tang.       M. 


1310 


403 


216 

430 


TABLE    XIII.       LOGrARlTHMlC    SINES, 


1370 


M. 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
23 
29 

30 
31 
i  32 
33 
34 
35 
3f3 
37 
3S 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
51 
55 
56 
57 
53 
59 
60 

M. 


i3a^ 


Sine. 


9.825511 
.825651 
.825791 
.825931 
.826071 
.826211 
.826351 
.826491 
.826631 
.826770 

9.826910 
.827049 
.827189 
.827328 
.827467 
.827606 
.827745 
.827384 
.823023 
.823162 

9.S2S301 
.828439 
.823578 
.823716 
.828855 
.828993 
.829131 
.829269 
.829407 
.829.545 

9.S29633 
.829321 
.829959 
.830097 
.830234 
830372 
.830509 
.830646 
.83)784 
.8.30921 

9. 33!  058 
.831195 
.831.332 
.831469 
.831606 
.831742 
.831879 
.832015 
.832152 
.832233 

9.832425 
.832561 
.832697 
.832333 
.832969 
.8.33105 
.833241 
.833377 
.833512 
.833643 
.833783 

Cosine. 


D.  v. 


2.34 
2.31 
2.33 
2.33 
2.33 
2.33 
2.-33 
2.-33 
2.-33 
2.33 

2.32 
2.32 
2.32 
2.32 
2.-32 
2.32 
2.-32 
2.31 
2.31 
2.31 

2.31 
2.31 
2.31 
2.31 
2.31 
2.30 
2.30 
2.30 
2.30 
2.30 

2.30 
2.30 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 
2.29 

2.23 
2.23 
2.23 
2.23 
2.23 
2.23 
2.23 
2.27 
2.27 
2.27 

2.27 
2.27 
2.27 
2.27 
2.27 
2.26 
2.26 
2.26 
2.26 
2.26 

D.  1". 


Cosine. 


9.871073 
.870960 
.870346 
.870732 
.870613 
.870504 
.870390 
.870276 
.870161 
.870047 

9.869933 
.869818 
.869704 
.869539 
.869474 
.869-360 
.869245 
.8691-30 
.869015 
.863900 

9-863735 
.863670 
.868555 
.863440 
.863324 
.863209 
.863093 
.867978 
.867862 
.867747 

9.867631 
.867515 
.867399 
.867233 
.867167 
.867051 
.8669-35 
.866319 
.866703 
.866586 

9.866470 
.866353 
.866237 
.866120 
.866004 
.865387 
.865770 
.86-5653 
.86.5536 
.86-5419 

9-86-5302 
.865185 
.865063 
.8&4950 
.864333 
.864716 
.864-593 
.864431 
.864363 
.864245 
.864127 

Sine. 


D.  1". 


1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.91 
1.91 

1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.91 
1.92 
1.92 
1.92 

1.92 
1.92 
1.92 
1.92 
1.92 
1.92 
1.93 
1.93 
1.93 
1.83 

1.93 
1.93 
1.93 
1.93 
1.93 
1.94 
1.94 
1.94 
1.94 
1.94 

1.94 
1.94 
1.94 
1  94 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 

1.95 
1.95 
1.95 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 
1.96 

D.  1". 


Tang. 


9.9-54437 
.9-54691 
.9.54946 
.9-55200 
-9554-54 
.955703 
.9-55961 
.9-56215 
.956469 
.9-56723 

9.956977 
.957231 
.957435 
.957739 
.957993 
.953247 
.953500 
.953754 
.959003 
.959262 

9.9-59516 
.9-59769 
.960023 
.960277 
.960530 
.960784 
.961033 
.961292 
.961545 
.961799 

9.9620-52 
.962306 
.962560 
.962313 
.963067 
.963320 
.963574 
.963323 
.964031 
.964335 

9-964583 
.964342 
.96.5095 
.965349 
.965602 
.965355 
.966109 
.966362 
.9666.6 
.966369 

9.967123 
.967376 
.967629 
.967333 
.963136 
.963339 
.963643 
.963396 
.969149 
.969403 
.969656 

Cotang. 


D.  1". 


4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 
4.23 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

D.  1'. 


Cotang. 


0.045563 
.045309 
.04.5054 
.044800 
.044546 
.044292 
.044039 
.0^43735 
.043531 
.043277 

0.04.3023 
.042769 
.042515 
.042261 
.042007 
.041753 
.041.500 

•  .041246 
.040992 


0.040484 

41 
40 

.040231 

39 

.039977 

33 

.039723 

37 

.039470 

36 

.039216 

35 

.033962 

34 

.038703 

33 

.033455 

32 

.033201 

31 

0.037943 

30 

.037694 

29 

.037440 

23 

.037187 

27 

.036933 

26 

.036630 

25 

.036426 

24 

.036172 

23 

.035919 

22 

.035665 

21 

0.03-5412 

20 

.035153 

19 

.034905 

18 

.034651 

17 

,034393 

16 

.0-341-15 

15 

.0-33391 

14 

.033633 

13 

.033334 

12 

.033131 

11 

0-032377 

10 

.032624 

9 

.0.32371 

8 

.032117 

7 

.031864 

6 

.031611 

5 

.0313-57 

4 

.031104 

3 

.0-30351 

2 

.030597 

1 

.030344 

0 
M 

Tang 

47' 


COSINES,    TANGENTS,    AND    COTANGENTS. 


430 


2n 


M. 

0 
1 
2 
3 
4 
5 

e 

7 


Sine. 


10 
11 
12 
13 
14 
15 
16 
17 
13 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
3S 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
GO 


D.  1". 


9.833783 
.833919 
.834054 
.834189 
.834325 
.834460 
.831595 
.834730 
.834365 
.834999 

9.835134 
.835269 
.835403 
.835r,.3S 
.8356:2 
.835807 
.835941 
.836075 
.836209 
.836313 

9.836477 
836611 
.836745 
.836378 
.837012 
.837146 
.837279 
.837412 
.837546 
,837679 

9.837812 
.837945 

.833078 
.833211 
.833344 
.833477 
.833610 
.833742 
.833375 
.839007 

9.839140 
.839272 
.839404 
.839536 
.839663 
.839800 
.839932 
.840064 
.840196 
.840323 

9.840459 
.840591 
.840722 
.840854 
.840985 
.841116 
.841247 
.841373 
.841509 
.841640 
.&il771 


M. 


Cosine. 


Cosine. 


2.26 
2.26 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 
2.25 

2.24 
2.24 
2.24 
2.24 
2.24 
2.24 
2.24 
2.23 
2.23 
2.23 

2.23 
2.23 
2.23 
2.23 

2.23 
2.22 
2.22 
2.22 
2.22 
2.22 

2.22 
2.22 
2.22 
2.21 
2.21 
2.21 
2.21 
2.21 
2.21 
2.21 

2.21 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.19 
2.19 

2.19 
2.19 
2.19 
2.19 
2.19 
2.19 
2.18 
2.18 
2.18 
2.18 


D.  1". 


9.864127 
.864010 
.863392 
.863774 
.863656 
.863533 
.863419 
.863301 
.863183 
.863064 

9.862946 
.862827 
.S627(.9 
.662590 
.862471 
.862353 
.862234 
.862115 
.861996 
.861877 

9.861758 
.861638 
^61519 
.861400 
.861230 
,861161 
.861041 
,860922 
,860302 
.860632 

9.860562 
.860442 
.860322 
.860202 
.860082 
.859962 
.859842 
.859721 
.859601 
.859480 

9.859360 
.859239 
.859119 
,858998 
.853877 
,858756 
.858635 
.858514 
.858393 
,858272 

9.858151 
.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857173 
.857056 
.856934 


D.  1". 


Tang. 


1.96 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 
1.97 

1.93 
1.93 
1.93 
1.93 
1.98 
1.98 
1.98 
1.98 
1.98 
1.99 

1.99 
1.99 
1.99 
1.99 
1.99 
1.99 
1.99 
2.00 
2.00 
2.00 

2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.01 
2.01 
2.01 
2.01 

2.01 
2.01 
2.01 
2.01 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 

2.02 
2.02 
2.02 
2.03 
2.03 
2.03 
2.03 
2.03 
2.03 
2.03 


Sine. 


D.  1". 


9.969656 
.969909 
.970162 
.970416 
.970669 
.970922 
.971175 
.971429 
.971682 
,971935 

9.972188 
,972441 
.972695 
.972943 
.973201 
.973454 
.973707 
.973960 
.974213 
,974466 

9.974720 
,974973 
,975226 
,975479 
.975732 
.975935 
.976233 
.976491 
.976744 
.976997 

9.977250 
,977503 
,977756 
,978009 
.978262 
.978515 
.978763 
.979021 
.979274 
.979527 

9.979730 
.980033 
,980286 
.980533 
.980791 
.981044 
.981297 
,981550 
,981803 
,932056 

9.982309 
.9S2562 
.932314 
.933067 
.983320 
.933573 
.983326 
.934079 
.984332 
.984534 
.934337 


D.  1".  Cotang. 


4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.21 
4.21 
4.21 
4.21 
4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 


Cotang. 


0.030344 
.030091 
.029833 
.029584 
.029331 
.029078 
.028825 
.028571 
.028318 
.028065 

0.027812 
.027559 
,027305 
,027052 
,026799 
.026546 
.026293 
.026040 
.025787 
.025534 

0.025280 
.025027 
.024774 
,024521 
.024263 
,024015 
,023762 
.023509 
.023256 
.023003 

0.022750 
.022497 
.022241  ■ 
.021991 
.021738 
.021435 
.021232 
.020979 
.020726 
.020473 

0.020220 
.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
,018450 
.018197 
,017944 

0.017691 
,017438 
,017186 
,016933 
.0166-^0 
,016427 
,016174 
.015921 
.015663 
.015416 
.015163 


M. 

60 

59 
58 
57 
56 
55 
54 


D.  1". 


Taug. 


50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

M. 


1 33  J 


46C 


218 

440 


TABLE    XIII. 


LOGARITHMIC    SINES,    &C. 


1354 


M. 

0 
I 
2 
3 
4 
5 
6 
7 
8 
9 

10 
II 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 

30 
31 
32 
33 
34 
35 
36 
37 
33 
39 

40 
41 
42 
43 
44 
45 
46 
47 
43 
49 

50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 


Sine. 


D.  1". 


9.34]  771 
.841902 
.842033 
.842163 
.842294 
.842424 

"  .842555 
.842635 
.842315 
.842946 

9.843076 
.843206 
.843336 
.84.3466 
.843595 
.843725 
.843355 
.843934 
.844114 
.844243 

9.844372 
.844502 
.844631 
.844760 
.844889 
.845018 
.845147 
.845276 
.84.5405 
.84.55.33 

9.845662 
.845790 
.84.5919 
.846047 
.846175 
.846304 
.846432 
.846558 
.846638 
.846316 

9.346944 
.847071 
.547199 
.847327 
.3474.54 
.847532 
.847709 
.847836 
.347964 
.843091 

9.848213 
.843345 

.843472 
.843599 
.813726 
.843352 
.843979 
.849106 
.8492-32 
.849.359 
.849435 


2.13 
2.18 
2  18 
2.13 
2.17 
2.17 
2.17 
2.17 
2.17 
2.17 

2.17 
2.17 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 
2.16 

2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.14 
2.14 

2.14 
2.14 
2.14 
2.14 
2.14 
2.14 
2.13 
2.13 
2.13 
2.13 

2.13 
2.13 
2.13 
2.13 
2.12 
2.12 
2.12 
2.12 
2.12 
2.12 

2.12 
2.12 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 
2.11 


Cosine. 


9.3.56934 
.856312 
.856690 
.856-568 
.856446 
.856323 
.856201 
.8.56078 
.85.59.56 
.855333 

9.8.5.5711 

.85.5533 
.855465 
.855342 
.8-55219 
.855096 
.8.54973 
.8543.50 
.8.54727 
.854603 

9.8.54430 
.8.543-56 
.854233 
.8-54109 
.8.53936 
.853362 
.8.53733 
.853614 
.853490 
.853366 

9.35-3242 
.8-53118 
.852994 
.352369 
.352745 
.852620 
.852496 
.852371 
.852247 
.852122 

9.851997 
.851372 
.851747 
.851622 
.851497 
.851372 
.851246 
.851121 
.850996 
.850370 

9.850745 
.850619 
.3.50493 
.850363 
.8-50242 
.8.50116 
.849990 
.849364 
.849733 
.849611 
.849435 


D.  1". 


M.  I    Cosine.    I   D.  1".   |     Sine.         D.  1".      Cotang.   |   D.  1". 


2.03 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 
2.04 

2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.06 
2.06 

2.06 
2.06 
2.06 
2.06 
2.06 
2.06 
2.06 
2.07 
2.07 
2.07 

2.07 

2.07 
2.07 
2.07 
2.07 
2.03 
2.03 
2.03 
2.03 
2.03 

2.03 
2.03 
2.03 
2.09 
2.09 
2.09 
2.09 
2.09 
2.09 
2.09 

2.09 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.10 
2.11 


Tang. 


9.934337 
.985090 
.935343 
.93.5.596 
.935343 
.936101 
.936-3-54 
.936607 
.936360 
.937112 

9.937365 
.937618 
.987871 
.938123 
.933376 
.933629 
.933332 
.939134 
.939337 
.939640 

9.939393 
.990145 
.990393 
.990551 
.990903 
.9911.56 
.991409 
.991662 
.991914 
.992167 

9.992420 
.992672 
.992925 
.993178 
.993431 
.99.3633 
.9939:36 
.994139 
.994441 
.994694 

9.994947 
.995199 
.995452 
.995705 
.995957 
.996210 
.996463 
.996715 
.996963 
.997221 

9.997473 
.997726 
.997979 
.993231 
.9934*4 
.993737 
.993939 
.999242 
.999495 
.999747 

0.000000 


D.  1". 


4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

4.21 
4.2) 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4.21 

4.21 
4.21 
4.21 
4.21 
4.21 
4.21 
4-21 
4.21 
4.21 
4.21 


Cotang. 


0.015163 
.014910 
.0146-57 
.014404 
.0141.52 
.013399 
.013646 
.013393 
.013140 
.012388 

0.012635 
.0123S2 
.012129 
.011877 
.011624 
.011.371 
.011118 
.010366 
.010613 
.010360 

0.010107 
.009355 
.009602 
.009349 
.009097 
.003844 
.003591 
.003333 
.003036 
.007833 

0.007530 
.007323 
.007075 
.006322 
.006569 
.006317 
.006064 
.005811 
.00.5559 
.005306 

0.005053 
.004301 
.004.543 
.004295 
.004043 
.003790 
.003537 
.003235 
.003032 
.002779 

0.002.527 
.002274 
.002021 
.001769 
.001516 
.001263 
.001011 
.000758 
.000.505 
.000253 
.000000 


Tang. 


M. 

60 
59 
53 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 
16 
15 
14 
13 
12 
II 

10 
9 
8 
7 
6 
5 
4 
3 
2 

I 
0 

M. 


134a 


i'i 


TABLE    XIV. 


NATURAL    SINES    AND    COSINES 


i-^{} 

TABLE 

XIV. 

NATURAL  SINES  AND  COSINES. 

M. 

0 

, 



= — il 

00 

i^ 

}c~^          \           a^ 

* 

^ 

1 
M. 

60 

Sine. 
.00000 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

.99756 

One. 

.01745 

.99985 

.03490 

.99939 

.0.5234 

.99863 

.06976 

1 

.00029 

One. 

.01774 

.999S4 

.03519 

.99933 

.05263 

.99-61 

.07005 

.99754 

59 

2 

.00058 

One. 

.01303 

.99934 

.03-543 

.99937 

.0.5292 

.99360 

.07034 

.99752 

58 

3 

.0JJS7 

One. 

.01332 

.99933 

.03577 

.99936 

05321 

.99358 

.07063 

.99750 

57 

4 

.00116 

One. 

.01362 

.99933 

.03606 

.99935 

.05-350 

.99357 

.07092 

.99748 

56 

5 

.00145 

One. 

.01391 

.99932 

.03635 

.99934 

05379 

.99355 

.07121 

.99746 

55 

6 

.00175 

One. 

.01920 

.99932 

.C3661 

.99933 

.0-5403 

.99354 

.07150 

.99744 

54 

7 

.00204 

One. 

.01919 

.99931 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742 

5c 

8 

.00233 

One. 

.01973 

.99930 

.03723 

.99931 

.0-5466 

.99351 

.07203 

.99740 

52 

9 

.00262 

One. 

.02097 

.99930 

.03752 

.99930 

.0.5495 

.99349 

.07237 

.99738 

51 

10 

.00291 

One. 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03310 

.99927 

.05553 

.99346 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.99973 

.0.38.39 

.99926 

.0-5532 

.99344 

.07324 

.99731 

48 

13 

.00373 

.99999 

.02123 

.99977 

.03363 

.99925 

.05611 

.99342 

.073.53 

.99729 

47 

14 

.00407 

.99999 

.021-52 

.99977 

.03397 

.99924 

.05640 

.99341 

.07332 

.99727 

46 

15 

.00436 

.99999 

.02131 

.99976 

.03926 

.99923 

.05669 

.993.39 

.07411 

.99725 

45 

16 

.00463 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99333 

.07440 

.99723 

44 

17 

.00495 

.99999 

.0221'! 

.99975 

.0-3934 

.99921 

.05727 

-99336 

.07469 

.99721 

43 

15 

.03524 

.99999 

.02269 

.99974 

.04013 

.99919 

.0-5756 

.99334 

.07493 

.99719 

42 

19 

.00553 

.99993 

.02293 

.99974 

.04042 

.99918 

.05785 

.998.33 

.07527 

.99716 

41 

20 

.00532 

.99993 

.02327 

.99973 

.04071 

.99917 

.0-5814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99993 

.02356 

.99372 

.04100 

.99916 

.05344 

.99329 

.07535 

.99712 

39 

22 

.00640 

.99993 

.02335 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

33 

23 

.03660 

.99993 

.02414 

.9p971 

.04159 

.99913 

.05902 

.99326 

.07643 

.99708 

37 

21 

.00693 

99993 

.02443 

.99970 

.04183 

.99912 

.05931 

.99-24 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99322 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02.501 

.99969 

.04246 

.99910 

.05989 

.99321 

.07733 

.99701 

^ 

27 

.00785 

.99997 

.02530 

.99963 

.04275 

.99909 

.06018 

.99319 

.07759 

.99699 

33 

2S 

.00314 

.99997 

.02.560 

.99967 

.04804 

.99907 

.06047 

.99317 

.07783 

.99696 

32 

29 

.00314 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

-99315 

.07817 

.99694 

SI 

30 

.O0S73 

.99996 

.02613 

.99966 

.04362 

.99995 

.06105 

.9981-3 

.07846 

.90692 

30 

31 

00502 

.99996 

.02647 

99965 

.04391 

.99904 

.061.34 

.99312 

.07875 

.9:^i 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.99902 

.06163 

.99310 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99333 

.07933 

.99635 

27 

34 

.009S9 

.99995 

.02734 

.99963 

.04478 

.99900 

.05221 

.99806 

.07962 

.99633 

26 

35 

.0101.3 

.99995 

.02763 

.99962 

.04.507 

.99393 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

.045.36 

.99397 

.06279 

.99-03 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02321 

.99960 

.04-565 

.99396 

.06303 

.99-01 

.08049 

.99676 

23 

3S 

.01105 

.99994 

.02350 

.99959 

.04594 

.99394 

.06337 

.99799 

.03073 

.99673 

22 

39 

.01134 

.99994 

.02379 

.999.59 

.04623 

.99393 

.06.566 

.99797 

.03107 

.99671 

21 

40 

.01164 

.99993 

.02903 

.99953 

.04653 

.99392 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.93993 

.02933 

.99957 

.04632 

.99390 

.06424 

.99793 

.03165 

.99666 

19 

42 

.01222 

.99993 

.02967 

.999.56 

.04711 

-99339 

.064-53 

.99792 

.08194 

.99664 

18 

43 

.01251 

.99992 

.02996 

.99955 

.04740 

.99333 

.06432 

.99790 

.03223 

-99661 

17 

44 

.01230 

.99992 

.03025 

.99954 

.04769 

.99336 

.06511 

.99788 

.08252 

.99659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99335 

.06540 

.99736 

.08281 

-99657 

15 

46 

.01.333 

.99991 

.0.3033 

.99952 

.04327 

.99333 

.06569 

.99734 

.08310 

-99654 

14 

47 

.0136/ 

.99991 

.03112 

.99952 

.048.56 

.99332 

.06598 

.99782 

.033.39 

.99652 

13 

4S 

.01396 

.99990 

.03141 

.99951 

.04335 

.99331 

.(,'6627 

.99780 

.03.363 

.99649 

12 

49 

.01425 

.99930 

.03170 

.99950 

.04914 

.99379 

.1)66-56 

.99773 

.03397 

.99647 

11 

50 

.01454 

.99939 

.03199 

.99949 

.04943 

.99-73 

.06635 

.99776 

-03426 

.99644 

10 

5[ 

.01433 

.99939 

.03223 

.99943 

.04972 

.99376 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99939 

.03257 

.99947 

.0-5001 

.99375 

."6743 

.99772 

.08434 

.996.39 

8 

53 

.01542 

.99933 

03236 

.99946 

.05030 

.99373 

.06773 

.99770 

.08513 

.99637 

/ 

54 

.01.571 

.99933 

.03316 

.99945 

.0.5059 

.99372 

.06-02 

.99763 

.03542 

.99635 

6 

55 

.01600 

.99937 

03345 

.99944 

.05083 

.99370 

M<31 

.99766 

.08.571 

.99632 

5 

56 

.01629 

.99937 

.03374 

.99943 

.05117 

.99369 

.06-^60 

.99764 

.08630 

.99630 

4 

57 

.01653 

.99936 

.0.3403 

.99942 

.05146 

.99-67 

.06339 

.99762 

.08629 

.99627 

3 

5^ 

.01637 

.99936 

.03432 

.99941 

.05175 

.99S6S 

.06918 

.99760 

.08653 

.99625 

2 

59 

.01716 

.99935 

.03461 

.99940 

.05205 

.99364 

.06947 

.997.53 

.08687 

.99622 

1 

60 
M. 

.01745 

.99935 

.03490 
Cosin. 

.99939 
Sine. 

.0-5234 

.99363 

-06976 

.997.56 
Sine. 

.08716 
Coein. 

.99619 
Sine. 

0 
M. 

Cosin. 

Sine. 

Cosin. 

Sine.  Cosin. 

8i 

P 

882    1 

87^    1    863 

85°    1 

TABLE    XTV.       x^ATURAL    SlIs'ES    AND    COSINES. 


221 


0 
I 
2 
3 


6 

7 

8 

9 
10 
11 
12 
13 
14 
15 

16 
17 
IS 
19 
20 
21 
22 
23 
24 
25 
26 
27 
2S 
29 
30 

31 
32 
33 
34 
35 
36 
37 
3S 
39 
40 
41 
42 
43 
44 
45 

l!) 

47 
4S 
49 
50 
51 
52 
53 
54 
55 
56 
57 
5S 
59 
60 

m7 


Sine-  Cosin. 

.ostTo' 

.0S745| 

.0S771 

.OSSO:} 

.0.SS31 

.OSSG  ) 

.03S^9 

.039 1 S 

.OS947 

.nS976 

.09)1)5 

.09[)3l 

.09063 

.09092 

.09121 

.091.50 


63 


.09179 
.092)55 
.092:^7 
.(I926G 
.09295 
.09321 
.09353 
.093S2 
.09411 
.09440 
.09469 
.0949S 
.09527 
.09556 
.09535 

.09614 
.09642 
.09671 
.09700 
.09729 
.0975S 
.097S7 
.09816 
.09345 
.09374 
.09903 
.09932 
.09961 
.09990 
.10019 


.99619 
.99617 
.99614 
.99612 
.99609 
.99607 
.99614 
.99602 
.99599 
.99596 
.99594 
.99591 
.9953  S 
.99536 
.9953  5 
.99530 

.99573 
.99575 
.99572 
.99570 
.99567 
.99564 
.99562 
.99559 
.99556 
.99553 
.99551 
.99543 
.99545 
.99542 
.99540 

.99537 
.99534 
.99531 
.99523 
.99526 
.99523 
.99520 
.99517 
.99514 
.99511 
.99503 
.99506 
.99503 
.99500 
.99497 


70 


8^ 


Siue.  Cosin.  |  Sine.  :  Cosin.  Sine.  Cosin 


.10043 
.10077 
.10106 
.10135 
.10164 
.10192 
.10221 
.10250 
.10279 
.10313 
.10337 
.10366 
.10395 
10424 
10453 


.99494- 
.99491 
.99433 
.99435 
.99432 
.99479 
.99476 
.99473 
.99470 
.99167 
.99461 
.99461 
.99453 
.99455 
.99452 


10453 

1043  i 

10511 

10'.4') 

10569 

10597 

10626 

10655 

10634 

10713 

10742 

.10771 

.1030: 1 

.10  52  J 

.10 553 

.10^37 

.10916 
.10945 
.10973 
.11002 
.11031 
.11060 
.11039 
.11113 
.11147 
.11176 
.11205 
.11234 
.11263 
.11291 
.11320 

.11349 
.11373 
.11407 
. 1 1436 
.11465 
.11494 
.11523 
.115.52 
.11530 
.11609 
.11633 
.11667 
.11696 
.11725 
.11754 


.99452 

.99119 
.99446 
.99143 
.99440 
.99437 
.99434 
.99131 
.99128 
.9.)l2l 
.99121 
.99113 
.99415 
.99112 
.99409 
.99406 

.99402 
.991H9 
.99396 
.9;)  !9  ! 
.99390 
.993>6 
.99333 
.99330 
.99377 
.99374 
.99370 
.99367 
.99364 
.99360 
.99357 

.99354 
.99351 
.99347 
.99314 
.99341 
.99337 
.99331 
.99331 
.99327 
.99324 
.99320 
.99317 
.99314 
.99310 
.99307 


Cosin.  Sine 

8lo 


11733 
11312 
1 1340 
11369 
11893 
.11927 
11956 
11935 
12914 
.12013 
.12071 
.12100 
.12129 
.121.53 
.12137 


.99303 
.99300 
.99297 
.99293 
.99299 
.99236 
.99233 
.99279 
.99276 
.99272 
.99269 
.99265 
.99262 
.99253 
.992.55 

Cosin.  Sine. 


833 


12137 

12216! 

12245 

12274 

I23i)2 
,12331 
.12369 
.12339 
.12413 
.12447 
.12476 
.12504 
.12533 
.12562 
.12591 
.12620 

.12619 
.12673 
.12706 
.12735 
.12764 
.12793 
.12322 
.12351 
.12330 
.12908 
.12937 
.12966 
12995 
.13024 
.13053 

.1303! 
.13110 
.13139 
.13163 
.13197 
.13226 
.1.3251 
1.3233 
,13312 
.13311 
.13370 
.1.3399 
.13427 
.134.56 
.13485 

.13514 
.13543 
.13.572 
.13600 
.13629 
.13653 
.13637 
.13716 
.13744 
.13773 
.13302 
13831 
.13360 
.13339 
.J3927 

Cosin.  Sine 

833" 


.99251 

.99243  , 

.99244  ' 

.99241)  ' 

.99237 

.99233 

.99230 

.99226, 

.99222 

.99219 

.99215 

.99211 

.99208 

.99204 

.99200 

.99197 
.99193 
.99139 
.99136 
.99132 
.99173 
.99175 
.99171 
.99167 
.99163 
.99160 
.99156 
.99152 
.99143 
.99144 

.99141 
.99137 
.99133 
.99129 
.99125 
.99122 
.99113 
.99114 
.99110 
.99106 
.99102 
.99098 
.99994 
.99091 
.99037 

.99033 
.99079 
.99075 
.99071 
.99067 
.99063 
.99059 
.99055 
.99051 
.99047 
.99043 
.99039 
.99035 
.99031 
.99027 


13917 
13946 
13975 
14i)'4 
14033 
14061 
11119) 
14119 
1414s 
,14177 
,14205 
,142.34 
.14  263 
.14292 
.14320 
.14349 

.14373 
.14407 
.14136 
.1446} 
.14493 
.14.522 
.14551 
.14.530 
.  14603 
.14637 
.14666 
.14695 
.14723 
.14752 
.14731 

.14310 

.14333 

.14367 

.14396 

14925 

14954 

14932 

1.5011 

1.5040 

1.5069 

,15097 

,15126 

,151.55 

,15134 

.15212 

,1.5241 
.15270 
.15299 
.15327 
.1.53.56 
.1.5335 
.15414 
.15442 
.15471 
15500 
.15529 
.155.57 
.15586 
.1.5615 
.15643 


90 


.99027 
.99023 
.99019 
.99015 
.99)11 
.990' )6 
.99002 
.93993 
.9>994 
.9>990 
.9 -•9 -6 
.9>9^2 
.93973 
.93973 
.93969 
.9^965 

.93961 

.93957 

.93953 

.9-^913 

.9^941 

.93940 

.9-936 

.93931 

.9>927 

.93923 

.93919 

.93914 

.93910 

.93906 

.939:12 

.93397 
.93393 
.93339 
.933S4 
.93330 
.93376 
.93871 
.9,3867 
.98363 
.93.353 
.988.54 
.98.349 
.93,345 
.93,341 
.93336 

.93332 
.93827 
.93823 
.9,3313 
.98314 
.93309 
.93305 
.93300 
.93796 
.93791 
.98737 
.987,32 
.9,8773 
.93773 
.93769 


Sine.  I  Cosin.  M 

J5643  .93769 
.1.5672  .9,8764 
.1.57111  .98760 
.15730  .93755 
.1575>  .93751 
.15737  .9.>746 
.1.5316  .9,3741 
.153451.93737 
.1.5373 '.93732 
.15902  .93723 
.1.5931  .93723 
.159.59  .93713 
.1.5933  .93714 
.16017  .93709 
.16046  .93704 
.16074  .93700 


Cosin.  Sine 

8I0 


16l!l3 
16132 
16160 
16139 
1621 
,16246 
,16275 
,16304 
.16333 
,16361 
,16.390 
,16419 
.16447 
.16476 
.16505 

.16533 
.16562 
.16591 
.16620 
.16643 
.16677 
.16706 
.167.34 
.16763 
.16792 
.16320 
.16349 
.16873 
.16906 
.16935 

.16964 
.16992 
.17021 
.17050 
.17073 
.17107 
,17136 
.17164 
.17193 
.17222 
.17250 
.17279 
.17.303 
.17336 
.17365 


60 
59 
53 
57 
56 
55 
54 
53 
52 
51 
50 
49 
43 
47 
46 
45 


44 
43 


93695 

93690 

9,3636  42 

.9,3631  41 

.93676  40 

.93671  39 

.95667 

.93602 

.98657 

.9,3652 

.9.3613 

.9,3643 

.936:33 

.93633 

.93629 


Cosin. 


93624 

93619 

93614 

93609 

98604 

93600 

9,359.' 

,9.8.590 

,9353i: 

,9,3580 

,93575 

.9,3570 

.93565 

,93561 

,93556 

.9,3551 

.93.546 

.93541 

.93536 

.93531 

.93.526 

.93521 

.93516 

.98511 

.93.506 

.98501 

.9,8496 

.93491 

.93436 

.93431 


38 
37 
36 
35 
34 
33 
32 
31 
30 


Sine. 


803 


29 
23 
27 
26 
25 
24 
23 
22 
21 
20 
!9 
13 
17 
16 
15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

J) 

M. 


222 


TABLE    XK 


i\ATU:iAL     bi:sKS    A.\D    COSINES. 


M. 

0 

103     1 

110 

130 

133 

140 

M. 

Sine. 
.17365 

Cosin. 

.93431 

Sine. 

Cosin. 
.93163 

Sine. 

Cosin. 

.97815 

Sine. 

Cosin. 
.97437 

Sine. 

Cosin. 

.19031 

.20791 

.22495 

.24192 

.970:30  60 

I 

.17393 

.93476 

.19109 

.93157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023  .59 

2 

.17422 

.93471 

.19133 

.93152 

.20348 

.97803 

.225.52 

.97424 

.24249 

.97015 

53 

3 

.17451 

.93466 

.19167 

.93146 

.20377 

.97797 

.22.530 

.97417 

.24277 

.97008 

57 

4 

.17479 

.93461 

.19195 

.93140 

.20905 

.97791 

.22603 

.97411 

.24305 

.97001 

56 

5 

.17503 

.93455 

.19224 

.931.35 

.20933 

.97734 

.22637 

.97404 

.24333 

.98994 

55 

6 

.17537 

.934.50 

.19252 

.93129 

.20962 

.97778 

.22665 

.97393 

.24-362 

.96987 

54 

7 

. 1 7565 

.93445 

.19231 

.98124 

.20990 

.97772 

.22693 

.97.391 

.24390 

.96980 

53 

8 

.17594 

.93440 

.19309 

.93113 

.21019 

.97766 

.22722 

.97334 

.24418 

.96973 

52 

9 

.17623 

.93435 

.193.33 

.93112 

.21047 

.97760 

.22750 

.97373 

.24446 

.96966 

51 

10 

.17651 

.93430 

.19-366 

.93107 

.21076 

.97754 

.22773 

.97371 

.24474 

.96959 

50 

11 

.17630 

.93425 

.19.395 

.9310! 

.21104 

.97748 

.22307 

.97.365 

.24503 

.96952 

49 

12 

.17703 

.93420 

.19123 

.93096 

21132 

.97742 

.22335 

.97.3-58 

.21531 

.96945 

48 

13 

.17737 

.93414 

.19452 

.93090 

.21161 

.97735 

.22363 

.97351 

.24559 

.96937 

47 

14 

.17766 

.93409 

.19431 

.93034 

.21189 

.97729 

.22^92 

97345 

.24587 

.96930 

46 

15 

.17794 

.934  >1 

.19509 

.93079 

.21218 

.97723 

.22920 

.97:3:38 

.24615 

.96923 

45 

16 

.17323 

.98399 

.19533 

.93073 

.21246 

.97717 

.22948 

.97.331 

.24644 

.96916 

44 

U 

.173.52 

.9.3394 

.19.566 

.93067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

13 

.17330 

.93339 

.19.595 

.93;i0l 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.93333 

.19623 

.93056 

.21331 

.97693 

.230.33 

.97311 

.24723 

.96394 

41 

20 

.17937 

.93373 

.19652 

.93050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96387 

40 

21 

.17966 

.93373 

.19630 

.93044 

.21338 

.97636 

.23090 

.97293 

.24734 

.96880 

39 

22 

.17995 

.93363 

.19709 

.93039 

.21417 

.97630 

.23118 

.97291 

.24313 

.96373 

33 

23 

.13023 

.93362 

.19737 

.930-33 

.21445 

.97673 

.23146 

.97234 

.24841 

.96866 

37 

21 

.130.52 

.93357 

.19766 

.93027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.13031 

.93352 

.19791 

.93021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.13109 

.93347 

.19323 

.93016 

.215.30 

.976.55 

.2.3231 

.97264 

.24925 

.96844 

.34 

27 

.131.33 

.93341 

.19351 

.93010 

.21559 

.97648 

.23260 

.97257 

.24954 

.96337 

33 

2S 

.13166 

.98336 

.19330 

.93004 

.21587 

.97642 

.23233 

.97251 

.24982 

.96329 

32 

1 

29 

.13195 

.93.331 

.19903 

.97998 

.21616 

.976.36 

.23316 

.97244 

.25010 

.96322 

31 

30 

.13224 

.93.325 

.199-37 

.97992 

.21644 

.97630 

.23:345 

.972:37 

.25033 

.96315 

30 

31 

.132.52 

.93.320 

.19965 

.97937 

.21672 

.97623 

.2.3373 

.972.30 

.25066 

.96807 

29 

32 

.13231 

.93315 

.19994 

.97931 

.21701 

.97617 

.23401 

.97223 

.25094 

.96300 

28 

33 

.13309 

.93310 

.20022 

.97975 

.21729 

.97611 

.2-3429 

.97217 

.25122 

.96793 

27 

34 

.13333 

.93.301 

.20051 

.97969 

.21758 

.97604 

.2:34-58 

.97210 

.25151 

.96786 

26 

35 

.13367 

.9.3299 

.20079 

.97963 

.21786 

.97593 

.23136 

.97203 

.25179 

.96778 

25 

36 

.13:395 

.93294 

.21103 

.97953 

.21814 

.97592 

.2:3514 

.97196 

.25207 

.96771 

24 

37 

.13424 

.93238 

.201.36 

.97952 

.21343 

.97535 

.2.3-542 

.97189 

.2.52.35 

.96764 

23 

33 

.1^.52 

.93283 

.20165 

.97946 

.21871 

.97579 

.2:3571 

.97132 

.2.5263 

.96756 

22 

39 

.13431 

.93277 

.20193 

.97940 

.21899 

.97573 

.2:3.599 

.97176 

.2.5291 

.96749 

21 

40 

.18509 

.93272 

.20222 

.979.34 

.21923 

.97566 

.23627 

.97169 

.2.5320 

.96742 

20 

41 

.13.538 

.93267 

.20250 

.97923 

.219.56 

.97560 

.2:3656 

.97162 

.25348 

.96734 

19 

42 

.18567 

.93261 

.20279 

.97922 

.21935 

.97553 

.23G34 

.97155 

.25376 

.96727 

18 

43 

.13595 

.932.56 

.20307 

.97916 

.22013 

.97547 

.2:3712 

.97143 

.2.5404 

.96719 

17 

44 

.13624 

.93250 

.20.3.36 

.97910 

.22041 

.97541 

.23740 

.97141 

.2.5432 

.96712  16| 

45 

.13652 

.93245 

.20364 

.97905 

.22070 

.97534 

.23769 

.971.34 

.25460 

.96705 

15 

46 

.18631 

.93210 

.20393 

.97399 

.22093 

.97523 

.2.3797 

.97127 

.2-5483 

.96697 

14 

47 

.13710 

.93234 

.20421 

.97893 

.22126 

.97521 

.23325 

.97120 

.25516 

.96690 

13 

43 

.13733 

.93229 

.20450 

.97337 

.221.55 

.97515 

.2:3353 

.97113 

.2.5.545 

.96632 

12 

49 

.13767 

.93223 

.20478 

.97831 

.22183 

.97503 

2-3332 

.97106 

.25573 

.96675 

11 

50 

.18795 

.93213 

.20.507 

.97375 

.22212 

.97502 

.23910 

.97100 

.2-5601 

.96667 

10 

51 

.13824 

.93212 

.20.535 

.97869 

.22240 

.97496 

.2:3935 

.97093 

.25629 

.96660 

9 

52 

.188.52 

.93207 

.20.563 

.97863 

.22263 

.97439 

.2:3966 

.97036 

.25657 

.96653 

8 

53 

.18331 

.93201 

.20.592 

.97357 

.22297 

.97433 

.23995 

.97079 

.2.5635 

.96645 

7 

54 

.18910 

.93196 

.20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96633 

6 

55 

.18933 

.93190 

.20649 

.97345 

.22.353 

.97470 

.21051 

.97065 

.25741 

.96630 

5 

56 

.18967 

.98135 

.20677 

.97339 

.22332 

.97463 

.24079 

.9705S 

.25769 

.96623 

4 

57 

.18995 

.93179 

.20706 

.97833 

.22410 

.974.57 

.24103 

.97051 

.25793 

.96615 

3 

53 

.19024 

.93174 

.207.34 

.97827 

.22438 

.974.50 

.24136 

.97044 

.25826 

.96603 

2 

59 

.19052 

.93163 

.20763 

.97321 

.22467 

.97444 

.24164 

.97037 

.25854 

.96600 

1 

60 
M. 

.19031 

.93163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25832 

.96593 

0 
M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosir. 

Sine. 

Cosin. 

Sine. 

1 

793    1 

780    1 

770    1 

76C    1 

7S 

P 

1 

1 

TABLE 

XIV. 

NATURAL 

SIxNES  AND  COSINES. 

ft 
< 

wa 

M. 

0 

150    1 

163    1    170    1 

183    1 

19a    1 

60 

Sine. 

.253 S2 

Cosia. 

Sine. 

Ccsin.  Sine.  1 

Cosin. 

.95630 

Sine. 

Cosin. 
.95106 

Sine. 

.32557 

Cosin. 

.96593 

.27.564 

.96126 

.29237 

.30902 

,94552 

1 

.25910 

.96535 

.27592 

.96118 

.29265 

.95622 

-30929 

.95097 

.32584 

.94542 

59 

2 

.25933 

.96578 

.27620 

.96110 

.29293 

.95613 

.30957 

.95088 

.32612 

.94.533 

58 

3 

.25960 

.96.570 

.27648 

.96102 

.29321 

.9-5605 

.30985 

.9.5079 

.32639 

.94.523 

57 

4 

.25994 

.90.562 

.27676 

.96094 

.29343 

.9.5596 

.31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704 

.960S6 

.29376 

.95538 

.31040 

.95061 

.32694 

.94504 

55 

^  6 

.26050 

.96547 

.27731 

.90078 

.29404 

.95579 

.31063 

.9-5052 

.32722 

.94495 

54 

7 

.26079 

.96.540 

.27759 

.96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94435 

53 

8 

.26107 

.965.32 

.27787 

.96062 

.29460 

.9-5562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96.524 

.27815 

.90f)54 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32332 

.94457 

50 

11 

.26191 

.90.509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95006 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95523 

.31233 

.94997 

.32837 

.94438 

48 

13 

.26247 

.96494 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914 

.94423 

47 

14 

.26275 

.96436 

.279.55 

.96013 

.29626 

.95511 

.31239 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96179 

.27983 

.96005 

.290-54 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

.26331 

.90171 

.2301 1 

.95997 

.29032 

.95493 

.31344 

.94961 

.32997 

.94.399 

44 

17 

.26359 

.96463 

.23039 

.95939 

.29710 

.95435 

.31-372 

.949.52 

.3-3024 

.94390 

43 

13 

.263S7 

.96456 

.23067 

.9-5981 

.29737 

.95476 

.31399 

.94943 

.3.3051 

.94380 

42 

19 

.20415 

.96443 

.23095 

.95972 

.29765 

.9.5467 

.31427 

.94933 

.33079 

.94370 

41 

20 

.26443 

.96440 

.23123 

.9-5964 

.29793 

.9.54.59 

.31454 

.94924 

.33106 

.94.361 

40 

21 

.26471 

.95433 

.23150 

.95956 

.29821 

.954-50 

.31482 

.94915 

.33134 

.94-351 

39 

22 

.26500 

.96425 

.28178 

.9.5943 

.29349 

.9-5441 

.31510 

.94906 

.33161 

.94.342 

33 

23 

.26523 

.96417 

.23206 

.95940 

.29876 

.9.5433 

.31.537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.232.34 

.95931 

.29904 

95424 

.31565 

.94383 

.33216 

.94.322 

36 

25 

.26534 

.964  )2 

.23262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.23290 

.9.5915 

.29960 

.9-5407 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26610 

.963S6 

.23318 

.95907 

.29987 

.95398 

.31648 

.94860 

.33293 

.94293 

33 

2S 

.26603 

.96379 

.23346 

.9.5393 

.30015 

.9-5339 

.31675 

.94851 

.33326 

.94234 

32 

29 

.26696 

.96371 

.23374 

.9-5390 

.30043 

.95330 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96363 

.23402 

.95382 

.30071 

.95372 

.31730 

.94832 

.33381 

; 94264 

30 

31 

.26752 

.96355 

.28429 

.9.5374 

.30093 

.95363 

.31758 

.94823 

.33408 

.94254 

29 

32 

.26780 

.96347 

.23457 

.9-5365 

.30126 

.95354 

.31786 

.94814 

.33436 

.94245 

2.8 

33 

.26303 

.96340 

.23435 

.95357 

.30154 

.9.5345 

.31813 

.94805 

.3-3463 

.94235 

27 

34 

.26836 

.96332 

.23513 

.95849 

.30182 

.9-5337 

.31841 

.94795 

.3-3490 

.94225 

26 

35 

.26364 

.96324 

.28541 

.9-5341 

.30209 

.95.323 

.31868 

.94786 

.33518 

.94215 

25 

36 

.26392 

.96316 

.28-569 

.9.5332 

.30237 

.9-5319 

.31896 

.94777 

.33.545 

.94206 

24 

37 

.2692  J 

.96303 

.23597 

.95324 

.30265 

.95310 

.31923 

.94763 

.3.3573 

.94196 

23 

3S 

.26943 

.96301 

.23625 

.9-5316 

..30292 

.95301 

.31951 

.94753 

.33600 

.94186 

22 

39 

.26976 

.96293 

.23652 

.95807 

.30320 

.95293 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96235 

.23630 

.95799 

.30348 

.95234 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.23703 

.9.5791 

.30376 

.95275 

.32034 

.94730 

.33632 

.941.57 

19 

42 

.2706'! 

.96269 

.237.36 

.95732 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27033 

.96261 

.23764 

.95774 

.30431 

.95257 

..32039 

.94712 

.33737 

.94137 

17 

44 

.27116 

.96253 

.23792 

.95766 

.30459 

.95243 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28320 

.'^rjlDl 

.30436 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96233 

.23847 

.95749 

.30514 

.95231 

.32171 

.94634 

.33319 

.94108 

14 

47 

.27200 

.90230 

.23375 

.95740 

.30.542 

.9-5222 

.32199 

.94674 

.33346 

.94098 

13 

4S 

.27223 

.96222 

.23903 

.95732 

.30570 

.9-5213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.96214 

.23931 

.95724 

..30597 

.9.5204 

32254 

.94656 

.3-3901 

.94078 

11 

50 

.27234 

.96206 

.23959 

.9.5715 

.30625 

.95195 

.32232 

.94646 

.33929 

.94068 

10 

51 

.27312 

,96193 

.23937 

.95707 

.30653 

.95186 

.32.309 

.94637 

.33956 

.940-58 

9 

52 

.27340 

.96190 

.29015 

.95693 

.30630 

.95177 

.32.337 

.94627 

.3.39-3 

.94049 

8 

53 

.27363 

.96182 

.29042 

.9509) 

.30703 

.95163 

.32364 

.94618 

.3401 1 

.940,39 

7 

54 

.27.396 

.96174 

.29070 

.9-5631 

.307.36 

.951.59 

.32392 

.94609 

.34038 

.94029 

6 

55 

.27421 

.96166 

.29093 

.95673 

.30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452 

.96153 

.29126 

.9-5664 

..30791 

.95142 

.32447 

.94.590 

.34093 

.94009 

4 

57 

.27430 

.96150 

.29154 

.95656 

..30319 

.95133 

.32474 

.94580 

..34,120 

.93999 

3 

5S 

.27503 

.96142 

.29132 

.9.5617 

.30346 

.95124 

.32502 

.94571 

.34147 

.9.3989 

2 

59 

.27536 

.961.34 

.29209 

.95639 

.SO  574 

.95115 

.32.529 

.94.561 

.34175 

.93979 

1 

60 
M. 

.27.561 
Cosin. 

.96126 
Sine. 

.29237 
Cosin. 

.95630 
Sine. 

.30902 
Cosin. 

.95106 

.32557 

.94.552 

.34202 
Cosin. 

.93969 
Sine. 

0 
M. 

Sine. 

Cosin. 

Sine. 

7 

40 

730        733    1    710 

703 

5^24 


TABLE    XIV, 


.NATURAL    Sl^'ES    AND    COSINES. 


M. 

0 

303 

310 

233 

333 

34:3 



Sine. 

.31211  > 

1  Cosin. 

Sine. 

.358:37 

Cosin. 

Sine. 

i  Cosin. 

Sine. 

Cosin. 

.92050 

Sine. 

Cosin. 
1.91355 

M. 

.9.3^6  J 

.9:3338 

.37461 

I.9271> 

.-39LI73 

.40674 

60 

1 

.34229 

!  .93959 

.35564 

.93:348 

.37483 

1.92707 

-39100 

.92039 

.40701 1 

.91343 

59 

2 

.34257 

1 .93949 

.35891 

.93:337 

.-37515 

1.92697 

-39127 

.92028 

.40727 

.91:331 

58 

3 

.:342>4 

i.9393D 

.3.59l> 

.93:327 

.37.542 

.926-6 

-.391-53 

.92016 

.40753 

.91319 

57 

4 

34311 

1.93929 

.33945 

.93316 

.37.569 

.92675 

.:39180 

.92005 

.40780 

.91:307 

.^6 

5 

.34339 

.93919 

.-35973 

.933;»6 

.37.595 

.92664 

.39207 

.91994 

.40=06 

.91295 

55 

6 

..34366 

.93909 

.36000 

.9-3295 

.:37622 

.92653 

.:392:34 

.919-2 

.40-33 

.91283 

54 

7 

.31-393 

.93899 

.36027 

.9:32-5 

.:37649 

.92642 

.:-926:) 

.91971 

.40560 

.91272 

53 

8 

.34421 

.93889 

..360.54 

.93274 

.37676 

.92631 

.3;J257 

.919:59 

.40886 

.91260 

52 

9 

.34445 

.93879 

.36051 

.93264 

.37703 

.9262  ) 

.39314 

.91948 

.40913 

.91248 

51 

10 

.34475 

.93?6J 

.3610- 

.932-53 

.37731 

.92609 

.3;«41 

.919:36 

.409-39 

.912.36 

50 

11 

.34503 

.93559 

.-36135 

.9:3243 

.37757 

.92598 

.:39.367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93549 

.:36162 

.932:52 

.37784 

.92587 

.39.394 

.91914 

.40992 

.91212 

4S 

13 

.34357 

.93539 

..36190 

.93222 

.:378ll 

.92576 

.:39421 

.91902 

.41019 

.9120(» 

17 

14 

.345S4 

.93529 

.:35217 

.9321  [ 

.37-35 

.92-365 

.3944- 

.91891 

.41045 

.9118> 

46 

15 

.31612 

.93519 

.:36244 

.93201 

.37865 

.925:54 

.-39474 

.91879 

.41072 

.91176 

45 

16 

.34639 

.93-09 

.:3627l 

.93190 

.37892 

.92.543 

.-39501 

.9186= 

.4109= 

.91164 

44 

17 

.34666 

.937^9 

.:36298 

.93150 

.37919 

.92.3:32 

.:3952? 

.918:36 

.41125 

.91152 

43 

IS 

.34694 

.937->>9 

.36:325 

.9316J 

.37946 

.92321 

.-39553 

.91845 

.41151 

.91140 

42 

19 

.34721 

.93779 

..36:332 

.93159 

.37973 

.92510 

.:39-38i 

.918:33 

.41178 

.91128 

41 

20 

.3474S 

.93769 

.-36379 

.93148 

.37999 

.92499 

.39605 

.91822 

.41204 

.91116 

40 

21 

.34775 

.937.59 

.:36106 

.931:37 

.35026 

.92455 

.39635 

.91810 

.412:31 

.91104 

.39 

22 

.34S03 

.93748 

.:364  34 

.93127 

.33053 

.92477 

.-39661 

.91799 

,412.57 

.91092 

38 

23 

.;MS3) 

.9373-^ 

.:334G1 

.93116 

.3305<i 

.92466 

.39688 

.91787 

.41254 

.91080 

37 

1 

21 

.:34S37 

.93728 

.:3645- 

.931(6 

..38107 

.924-35 

..39715 

.91775 

.41310 

.9106= 

36 

23 

.34SS4 

.93718 

.36515 

.9.3095 

.3-134 

.92444 

.:39741 

.917fr4 

.413:37 

.910:36 

35 

26 

.34912 

.93708 

.:365I2 

.9:3084 

.351Gi 

.924:32 

.:39768 

.91752 

.41.363 

.91044 

:34 

27 

.3193.^ 

.93695 

.-365^15 

.9:3074 

.3^188 

.92421 

.:39795 

.91741 

.41390 

.910.32 

.33 

2S 

.34956 

.93658 

.36536 

.9:3!')63 

.:38215 

.92410 

..39-22 

.91729 

.41416 

.91020 

32 

29 

.34993 

.93677 

;36'323 

.93052 

.:33241 

.92:399 

..39848 

.91718 

.41443 

.91008 

31 

30 

.35021 

.93667 

.:36650 

.9.3042 

.:38265 

.92358 

.:39875 

.91706 

.41469 

.90996 

30 

31 

.3504- 

.93657 

.36677 

.93031 

.3^^293 

.92.377 

.39902 

.91694 

.41496 

.90934 

29 

' 

32 

.33075 

.93647 

.337  '4 

.9:3020 

.3 -322 

.92366 

.-3992? 

.916=3 

.41,-22 

.90972 

2= 

33 

.33102 

.9:;637 

.3573! 

.9:3:>:il 

.3?:349 

.92:355 

.:39955 

.91671 

.41549 

.90960 

27 

34 

.:!5I30 

.93526 

.36758 

.92999 

.3<3:6 

.92:343 

.39982 

.91660 

.41.575 

.90948 

26 

i 

33 

.331-37 

.93616 

.337<3 

.92955 

.3-403 

.92:3-32 

.40005 

.9164- 

.41602 

.909-36 

25 

1 

3n 

.33I>4 

.9:3606 

.:3681  i 

.92978 

.384:^0 

.92:321 

.40035 

.916:36 

.4162= 

.90924 

24 

37 

.33211 

.93596 

.365:3H 

.92967 

.:38156 

.92310 

.40062 

.91625 

.41655 

.9091 1 

23 

3i 

.35230 

.9:3535 

.:36567 

.92956 

.3?483 

.92299 

.40083 

.91613 

.41651 

.90899 

22 

39 

.-35266 

.9:3-575 

.:36594 

.92915 

.35510 

.92287 

.40115 

.91601 

.41707 

.90337 

21 

40 

.35293 

.93565 

.:36921 

.929:J5 

.:38537 

.92276 

.40141 

.91-590 

.4173-1 

.90375 

20 

41 

.35320 

.9:3355 

.3394N 

.92924 

.38.564 

.92265 

.4016= 

.91:373 

.41760 

.90363 

19 

42 

.35:347 

.9:3.544 

..36975 

.9291:^ 

.35.591 

.922.54 

.40195 

.91566 

.41737 

.90=51 

18 

43 

.35375 

.9.3->34 

.37002 

.92902 

.35617 

.92243 

.40221 

.91:555 

.41313 

.90839] 

17 

• 

44 

..35402 

.93-524 

.37029 

.92-92 

.35644 

.92231 

.40243 

.91.543 

.41840 

.90826  16 

45 

.35429 

.9:3514 

.370.56 

.92881 

.:35671 

.92220 

.40275 

.91-531 

.41866 

.908141  15 

46 

..35456 

.93503 

.37033 

.92-70 

.35698 

.92209 

.40.301 

.91519 

.41892 

.903021 

14 

47 

.3:5454 

.93493 

.37110 

.92559 

.38725 

.92193 

.40:325 

.91:31)8 

.41919 

.90790! 

13 

4S 

..35511 

.93453 

.:371.37 

.92849 

.337.52 

.92186 

.40:3.55 

.91496 

.41945 

.90773! 

12 

49 

.3553S 

.93472 

.37164 

.92-38 

.38778 

.92175 

4J:381 

.914=4 

.41972 

.907661 

11 

50 

.33565 

.93462 

.-37191 

.92-27 

.:38805 

.92164 

.40408 

.91472 

.41998 

.907.53 

10 

51 

.33592 

.93452 

.37215 

.92.316 

-388-32 

.92152 

.4114.34 

.91461 

.42024 

.90741 

9 

52 

.35619 

.9:3411 

.37245 

.92805 

-335.59 

.92141 

.40461 

.91449 

.42051 

.90729 

8 

53 

.33647 

.93431 

.37272 

.92794 

.38886 

.92130 

.40488 

.914:37 

.42077 

.9(J717 

7 

54 

.-35674 

.93420 

.37299 

.92754 

.33912 

.92119 

.40514 

.91425 

.42104 

.90704 

6 

55 

.35701 

.93410 

.37.326 

.92773 

.33939 

.92107 

.40541 

.91414 

.421.30 

.90692 

5 

56 

.3572S 

.93400 

.37353 

.92762 

.33966 

.92096 

.40567 

.91402 

.421:56 

.90680 

4 

57 

.■ir>/00 

.a3389 

.37:380 

.92751 

.-38993 

.92055 

.40594 

.91390 

.42133 

.90663 

3 

53 1 

.35782 

.93379 

.37407 

.92740 

.39020 

.92073 

.4)621 

.91378 

.42209 

.9CK555 

2 

59 

.35810 

.9-3-363 

..374.34 

.92729 

.39046 

.92062 

.40647 

.91.366 

.422:35 

.£0643 

I 

60 ; 

M.  j 

1 

.35837 

.933-58 
Sine. 

.:37461 
Cosin. 

.9271- 
Sine. 

.3£073 
Cosin. 

.920-50 

.40674 

.9ia35 
Sine. 

.42262 
Cosin. 

.90631 
Sine. 

0 
M. 

Cosin. 

Sine. 

Cosin. 

693    1 

683    1 

673    1 

663    1 

653 

TABLE 

XIV. 

NATURAI 

.  SINES  AND  COSINES 

>. 

n 

M. 

0 

35^ 

3G3 

27- 

38-^ 

39^ 

M. 

60 

Sine. 
.42262 

Oosia. 

Sine. 

.43337 

Ccsin. 

■89379 

Sine. 

Gosin. 

Sine. 

Cosin. 

.88295 

Sine. 
.48481 

Cosin. 

.90631 

.43399 

.89101 

.46947 

.87462 

I 

A22^S 

.90618 

.43363 

.39S67 

.4.5425 

'  .89087 

.46973 

.88281 

.43506 

.87443 

59 

2 

.42315 

.90306 

.43389 

.39354 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

53 

3 

.42311 

.90594 

.43916 

.S9S41 

.45477 

.89061 

.47024 

.83254 

.48557 

.87420 

57 

4 

.42367 

.905S2 

.43942 

.89S23 

.45503 

.89048 

.47050 

.8321(1 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43963 

.89316 

.45529 

.890.35 

.47076 

.88226 

.43608 

.87.391 

55 

6 

.42120 

.90557 

.43991 

.89803 

.45.554 

.89021 

.47101 

.83213 

.43634 

.87377 

54 

7 

42446 

.90545 

.44020 

.89790 

.45530 

.89003 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42173 

.90532 

.44046 

.89777 

.45606 

.83995 

.47153 

.88185 

.48634 

.87349 

52 

9 

.42199 

.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44093 

.89752 

.45658 

.83968 

.47204 

.83158 

.48735 

.87321 

50 

11 

.42.552 

.90495 

.44124 

.89739 

.45634 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42573 

.90433 

.14151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88923 

.47231 

.88117 

.4881 1 

.87278 

47 

14 

.42631 

.90453 

.44203 

.89700 

.45762 

.88915 

.47306 

.83103 

.43837 

.87264 

46 

15 

.42657 

.90146 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42633 

.904.33 

.412.55 

.89674 

.45313 

.88883 

.47358 

.88075 

.48868 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45339 

.88375 

.47333 

.88(162 

.48913 

.87221 

43 

IS 

.4273''; 

.90403 

.44307 

.89619 

.45865 

.88862 

.47409 

.88043 

.43938 

.87207 

42 

19 

.42762 

.9:)396 

.44.333 

.89636 

.45891 

.83848 

.47431 

.830.34 

.43964 

.87193 

41 

20 

.427-:;S 

.90333 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42?15 

.90371 

.44335 

89610 

.45942 

.88822 

.47486 

.83006 

.49014 

.87161 

39 

22 

.42341 

.90358 

.44411 

.89.597 

.45963 

.83308 

.4751 1 

.87993 

.49040 

.87150 

33 

23 

.42S67 

90.346 

.44437 

.89584 

.45994 

.83795 

.47.537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

2.3 

.42920 

.90.321 

.44490 

.895.33 

.46046 

.83768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

23 

.42999 

.90234 

.44.368 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.902.59 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.4.3077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49263 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46226 

.83674 

.47767 

.878.54 

.49293 

.87007 

28 

33 

.431.30 

.90221 

.44693 

89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

43156 

.90208 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49.344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89423 

.46301 

.88634 

.47844 

.87812 

.49.369 

.86964 

25 

36 

.4.3209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47395 

.87784 

.49419 

.86935 

23 

3S 

.43261 

.90153 

.44323 

.89339 

.46381 

.83593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43237 

.90146 

.44354 

.89376 

.46407 

.83580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44380 

.89363 

.46433 

.88566 

.47971 

.87743 

49495 

.86392 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46453 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90103 

.44932 

.89337 

.46484 

.88.539 

.43022 

.87715 

.49.546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89321 

.46510 

.83526 

.43043 

.87701 

.49571 

.86349 

17 

44 

.43413 

.90082 

.44934 

.89311 

.465.36 

.83512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43145 

.90070 

.45010 

.89293 

.46561 

.88499 

.43099 

.87673 

.49622 

.86320 

15 

46 

.43471 

.90057 

.45036 

.89235 

.46587 

.83435 

.48124 

.87659 

.49647 

.86305 

14 

47 

.43197 

.90045 

.45052 

.89272 

.46613 

.83472 

.481.50 

.87645 

.49672 

.86791 

13 

4S 

.43523 

.90032 

.4:5038 

.89259 

.46639 

.88453 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.83431 

43226 

.87603 

.49743 

.86743 

10 

51 

.4.3602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43623 

.89931 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49793 

.86719 

8 

53 

.436.54 

.89963 

.45218 

.89193 

.46767 

.83390 

.48303 

.87.561 

.49824 

.86704 

7 

54 

.4.3630 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

,49349 

.86690 

6 

55 

.43706 

.89943 

.4.5269 

.89167 

.46319 

.83363 

.483.54 

.87532 

.49374 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.43379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

63 

.43785 

.899*5 

.4.5.347 

.89127 

.46896 

.83322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43311 

.89392 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43337 

.89379 
SineJ 

.45399 

.89101 

.46947 

.88295 
Sine. 

.48481 

.87462 
Sine. 

.50000 

.86603 

0 
M. 

Cosln. 

Cosln. 

Sine. 

Cosln. 

Cosin. 

Cosin. 

Sine. 

640   1 

030    1 

eao      I 

610    1 

603    1 

11 


226 


TABLE    XIV.       NATURAL    SINES    AND    COSINES. 


~0 

303    1 

310    1 

333    1 

333    1 

343 

M. 

60 

Sine. 
.50000 

Cosin. 

Si.:e. 

Co-iii. 

Slue. 

Ccsin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

.86603 

51504 

.85717 

.52992 

.84305 

.54464 

.3:3367 

.55919 

.>29J4 

I 

.50J25 

.86533 

51529 

.85702 

.53017 

.84789 

.544-5 

,83351 

.55943 

.82337 

59 

2 

.5  J  150 

.86573 

51554 

.85687 

.53041 

.84774 

.-,4513 

.83335 

.55963 

.82371 

58 

3 

.50076- 

.56559 

.51579 

.85(72 

.53066 

.^4759 

.54537 

.33319 

.55992 

.82355 

57 

-» 

50101 

.86544 

.51604 

.85657 

.53091 

.34743 

.54561 

.83304 

..56016 

.82539 

56 

.? 

.50126 

.86530 

.51623 

.-55642 

.53115 

.3472> 

.."^45-6 

.33788 

.56040 

,82322 

55 

6 

.50151 

.86515 

.51653 

.35627 

.53140 

.34712 

.54610 

.33772 

.56064 

,82306 

54 

7 

..50176 

.86501 

.51678 

.85612 

.53h:4 

.34fi97 

.546:35 

.83756 

.56033 

.82790 

53 

5 

.50201 

.86486 

.517(3 

.35597 

.53189 

.84631 

.54659 

.8:3740 

.56112 

.82773 

52 

9 

.50227 

.86471 

.51728 

.^55>i 

5.3214 

.54666 

,54653 

.33724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.'^."i-'ifl, 

.53,-38 

.34650 

..54703 

.8370- 

.56160 

.82741 

50 

11 

50277 

.86442 

.51778 

.85551 

.53263 

.^4635 

.=547.32 

.83692 

..56134 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.547.56 

.8.3676 

..56203 

,82708 

48 

Vo 

.50327 

.86413 

.51323 

.85521 

.53312 

.3 16;  4 

.54731 

.-366  1 

.562.32 

.82692 

47 

14 

.50352 

.8639S 

.513.52 

.855  16 

.53337 

.8153- 

.&4S05 

.3:3645 

.56256 

.82675 

46 

15 

.50377 

.86334 

.51377 

.854^; 

..53J61 

.84573 

.54829 

.3:3629 

.56280 

.82659 

45 

16 

.50403 

.86.369 

.51902 

.85476 

.53336 

.-^15.'>7 

.543.54 

.s;3613 

..56:3)5 

.82643 

44 

17 

.5042- 

.863.54 

.51927 

.8.5461 

..53411 

.34rv42 

.54? 78 

.83597 

.56:329 

.82626 

43 

IS 

.50453 

.86340 

.519.52 

.35416 

.m435 

.34 '.if; 

.54902 

.S3531 

.56353 

.82610 

42 

19 

.5047S 

.86325 

.51977 

.3)4:1 

.53460 

.84511 

.54927 

.83565 

.56377 

.82593 

41 

20 

.50503 

.8o310 

.52002 

.85!:  6 

.5.34>l 

.84495 

.54951 

'.3:3.549 

..56401 

.82577 

40 

21 

.5052S 

.8rt-2:'5 

.52026 

.3.5401 

.53509 

.814-0 

..54975 

.8:3533 

..56425 

.82561 

39 

22 

..50.5.53 

.86231 

..52)51 

.35335 

.53531 

.34l6i 

.54999 

.83517 

.56449 

.82544 

33 

23 

.50578 

.86266 

.52076 

.85370 

.53553 

.3411- 

.5.5024 

.83501 

.56473 

.82523 

37 

21 

.50603 

.862,31 

..52101 

.85355 

.53.533 

.34433 

.55043 

..S34-5 

.56497 

.82511 

36 

25 

..50623 

.86237 

.521  6 

.8534  . 

.53607 

.84417 

.5.5072 

.33469 

.56521 

.82495 

35 

26 

.506.54 

.862i2 

.52151 

.85325 

.53632 

.344' 2 

.55097 

.3-3453 

..56545 

.82473 

.34 

27 

.50679 

.86207 

..52175 

.85310 

..".3656 

.-<!!- 6 

.55121 

.Si43r 

.56569 

.82462 

33 

2> 

.50701 

.86192 

..52200 

.35294 

.53631 

.S4370 

.55145 

.8:3421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84:3.55 

.55169 

.3:3405 

.56617 

.82429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86143 

.52275 

.85249 

.537.->l 

.34.324 

.55218 

.33373 

.56665 

.82396 

29 

32 

..50304 

.86133 

.52299 

.3.5231 

.53779 

.34313 

.55242 

.83356 

.56639 

.82330 

23 

33 

.50329 

.86119 

.52321 

.8.5213 

.53>04 

.84292 

.5.5266 

.83340 

.56713 

.82363 

27 

3; 

.503:54 

.86101 

.5234:^ 

.85ai3 

.5332- 

.64277 

.5529[ 

.83:324 

.567.36 

.82:347 

26 

35 

.50379 

.86039 

.52374 

.35133 

.53353 

.84261 

.55315 

.83303 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52.399 

.s-)l7;i 

.53377 

.34245 

.55:3:39 

.83292 

.56784 

.82314 

24 

37 

..50929 

.86059 

.52423 

.85157 

.53902 

.^42:30 

.55363 

.33276 

.56303 

.82297 

23 

3> 

.50954 

.85045 

.52443 

.85142 

.53926 

.84214 

.55333 

.8.3260 

.563.32 

.82231 

22 

39 

.50979 

.85030 

.52473 

.35127 

.53951 

.81193 

.55412 

.3:3244 

.56356 

.82264 

21 

40 

.51004 

.86015 

.5243> 

.85112 

.53975 

.34132 

.5.5436 

.■^3223 

.56330 

.82243 

20 

41 

.51029 

.86000 

.52.522 

.85096 

.54000 

.81167 

.5.5460 

.8.32  U 

.56904 

.82231 

19 

42 

.51054 

.85935 

.52547 

.85081 

.54024 

.84151 

.55434 

:83195 

.56923 

.32214 

18 

43 

.51079 

.8597(1 

.52572 

.3.5006 

.54049 

.84135 

.5.5509 

.83179 

.569.52 

.8219? 

17 

44 

.51104 

.35956 

..52597 

.850.51 

.54073 

.84120 

.5.55.3.3 

.83163 

.56976 

.82181 

16 

45 

.51129 

.8594 1 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.57000 

.82165 

15 

46 

.511.54 

.85926 

..52646 

.35020 

.54122 

.84033 

..5.5.531 

.83131 

.57024 

.82143 

14 

47 

.51179 

.&5911 

..52671 

.85005 

.54146 

.84072 

..5.5605 

.83115 

.57047 

.82132 

13 

4S 

.51204 

.85396 

..52696 

.84939 

.54171 

.84057 

.556:30 

.83098 

.57071 

.32115 

12 

49 

.51229 

.85331 

..52720 

.34974 

.54195 

.84041 

..55654 

.83082 

.57095 

.82098 

11 

50 

.51254 

.35366 

.52745 

.84959 

..54220 

.34025 

.55678 

.3:3066 

.57119 

.82082 

10 

51 

.51279 

.3^5-51 

.52770 

.34943 

..54244 

.34009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85336 

.52794 

.8492? 

.54269 

.33994 

.55726 

.8.30.34 

.57167 

.82043 

8 

53 

.51329 

.85321 

..52319 

.34913 

..54293 

.83973 

.55750 

.83017 

.57191 

.820.32 

7 

54 

.51.354 

.85306 

..52344 

.■34397 

.54317 

.83962 

.  .55775 

.8.3001 

.57215 

.82015 

6 

55 

.51379 

.85792 

..52369 

.84332 

.54342 

.83946 

..55799 

.82935 

.572.33 

.81999 

5 

56 

.51404 

.85777 

..52393 

.84366 

.54366 

.839.30 

.55823 

.82969 

.57262 

.81932 

4 

57 

.51429 

.85762 

.52918 

.84351 

.54391 

.83915 

.5.5347 

.82953 

.57286 

.81965 

3 

53 

.51454 

.85747 

..52943 

.84836 

..54415 

.83399 

.5=5871 

.82936 

.57310 

.81&49 

2 

59 

.51479 

.85732 

.52967 

.84S20 

..54440 

.83383 

.55895 

.82920 

.57334 

.81932 

1 

60 
M. 

.51504 

.85717 
Sine. 

.52992 
Cosin. 

.84805 
Sine. 

.54464 
Coain. 

.83367 

.55919 

.82904 

.57353 

.81915 

0 
M. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

593    1 

5 

33 
'—^ 

573    < 

563 

553 

TABLE    XIV.       NATURAL    SINES    AND    COSINES. 


227 


M. 

C 

350 

3GO 

370    1    383 

390 

M. 

)  60 

Sine. 

.573.jt: 

Cosin 

Sine. 

Cosin. 

.80902 

Sine. 

Cosin. 

.79364 

Sine. 

Cosin. 

.78301 

Sine. 
.62935 

Cosin 

.777K 

.«1915 

.5S77<J 

.60185 

.6156e 

1 

.57381 

.81399 

.58302 

.80S8.') 

.6020- 

.79346 

.6153£ 

.78783 

.62955 

.7769t 

)  69 

2 

.57405 

.81382 

.5S32G 

.80367 

.6022.5 

.79329 

.61615 

.78765 

.62977 

.77676 

]   58 

3 

.57420 

.81865 

.58349 

.80350 

.60251 

.79311 

.61635 

.78747 

.63000 

.7766t 

)  57 

4 

.57453 

.81843 

.58873 

.80333 

.60274 

.79793  J  .61953 

.78729 

.63022 

.77641 

56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.7977G 

.61631 

.78711 

.G3045 

.77625 

55 

G 

.57501 

.81815 

.58920 

.80799 

.60.321 

.79758 

.61704 

.76694 

.630G6 

.7760£ 

54 

7 

Sural 

.81793 

.58943 

.8078-2 

.60344 

.79741 

.6172G 

.78676 

.6.3090 

.7758C 

53 

8 

.57:-i- 

.81782 

.58967 

.30765 

.G0367 

.79723 

.61749 

.78653 

.63113 

.77563 

52 

9 

.57572 

.81765 

.58990 

.80743 

.60390 

.79706 

.61772 

.78640 

.63135 

.7755C 

51 

10 

."j/o'.iG 

.81748 

.59014 

.80730 

.60414 

.79638 

.61795 

.78622 

.63158 

.77531 

50 

11 

.570 19 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77512 

49 

12 

.57613 

.81714 

.59061 

.80696 

.60460 

.796.53 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.78563 

.63225 

.77476 

47 

14 

.57691 

.81631  S.. 59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.62909 

.78532 

.63271 

.77439 

45 

16 

.57733 

.81647 

.59154 

.80627 

.60053 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

..59173 

.80610 

.60576 

.79.565 

.61955 

.78496 

.63316 

.77402 

43 

13 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63333 

.77384 

42 

19 

.57^10 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.764G0 

.63.361 

.77366 

41 

20 

.575.33 

.81580 

.59248 

.80553 

.60645 

.79512 

.62024 

.78442 

.63333 

.77347 

40 

21 

.57857 

.ai563 

.59272 

.80541 

.60G63 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63423 

.77310 

38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78337 

.6.3451 

.77292 

37 

24 

.57923 

.81513 

.59312 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273 

36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.772-55 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.81462 

.59412 

.80433 

.60807 

.79338 

.62183 

.78315 

.63540 

.77218 

33 

23 

.58023 

.81445 

.594.36 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

.53017 

.81423 

.59459 

.80403 

.60853 

.79353 

.62229 

.78279 

.63535 

.77181 

31 

30 

.53070 

.81412 

.59482 

.80336 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.53094 

.81395 

.59506 

.80363 

.60899 

.79318 

.62274 

.78243 

.6.3630 

.77144 

29 

32 

.53118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.76225 

.63653 

.77125 

28 

33 

.53141 

.81361 

.59552 

.803.34 

.60945 

.79232 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60963 

.79264 

.62.342 

.78158 

.63693 

.77088 

26 

35 

.53189 

.81327 

.59599 

.80299 

.60991 

.79247  .62365 

.78170 

.63720 

.77070 

25 

36 

.53212 

.81310 

.59622 

.80282 

.61015 

.79229  .62-3881 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61033 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

33 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.73093 

.63310 

.76996 

2! 

40 

.53307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.G3S32 

.76977, 

20 

41 

.533.30 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63354 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63377 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.6.3399 

.76921 

17 

44 

.53401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.73007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59332 

.80125 

.61222 

.79069 

.62.592 

.77983 

.6.3944 

.76884 

15 

46 

.53449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76366 

14 

47 

.53472 

.81123 

.59879 

.80091 

.61263 

.79033 

.62633 

.77952 

.639>9 

.76347 

13 

48 

.53496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77931 

.64011 

.7GS2S 

12 

49 

.53519 

.81089 

.59926 

.80056 

.61314 

.78998 

62633 

.77916 

.64033 

.76810 

11 

50 

..58543 

.81072 

.59949 

.80038 

.61337 

.78^30 

.62706 

.77897 

.64056 

76791 

10 

51 

.53567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

76772 

9 

52 

.58590 

.81038 

.59995 

.60003 

.61333 

.78944 

.62751 

.77661 

.64100 

76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

76735 

7 

54 

.586.37 

.81004 

.60042 

.79963 

.61429 

.78908 

.62796 

77824 

.64145 

76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62319 

77806 

.64167 

76698 

5 

56 

.58634 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

77788 

.64190 

76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

78855 

.62864 

77769 

.64212 

76661 

3 

58 

.58731 

.80936 

.60135 

.79399 

.61.520 

78837 

.62887 

77751 

.64234 

76642 

2 

59 

58755 

.80919 

.60158 

.79381 

.61543 

78819 

.62909 

77733 

.64256  . 

76623 

1 

60 
M. 

58779 
CJosin. 

80902 

.60182 

79864 

.61566 

78801 

.62932 
Cosin. 

77715 

.64279  . 
Cosin. 

76604 

0 

Sine.  Cosin.  1 

Sine. 

Cosin. 

Sine. 

Sire. 

Sine.  I 

540    1    530    1 

sao      r 

510    1 

500    1 

228 


TAB].E    XIV.       NATURAL    SINES    AND    COSINES. 


M. 

0 

4:03 

4:10 

4:30 

4:33 

4:40 

M. 

60 

Sine. 

.64279 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

Sine. 

Cosin. 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.719.S4 

1 

.64301 

.76586 

.65623 

.75452 

.66935 

.74295 

.68221 

.73116 

.69437 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69503 

.71894 

53 

3 

.64346 

.76543 

.65672 

.7.5414 

.66978 

.74256 

.68264 

.73076 

.69.529 

.71873 

57 

4 

64363 

.76.530 

.65694 

.75395 

.66999 

.74237 

.63235 

.73356 

.69549 

.71853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68.306 

.73036 

.69570 

.71833 

55 

6 

.64412 

.76492 

.65733 

.75.356 

.67043 

.74193 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74173 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65731 

.75313 

.67086 

.74159 

.68.370 

.72976 

.69633 

.7F72 

52 

9 

.64479 

.76436 

.65303 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.6582.5 

.75280 

.67129 

.7412) 

.63412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76393 

.6.5347 

.75261 

.67151 

.74100 

.63434 

.72917 

.69696 

.71711 

49 

12 

.64546 

.76330 

.65369 

.75241 

.67172 

.74080 

.63455 

.72897 

.69717 

.71691 

43 

13 

.64568 

.76361 

.65391 

.75222 

.67194 

.74061 

.63476 

.72377 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69753 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75134 

.67237 

.74022 

.63513 

.72837 

.69779 

.71630 

45 

16 

.646.35 

.76304 

.65956 

.75165 

.67258 

.74002 

.685.39 

.72317 

.69300 

.71610 

44' 

17 

.64657 

.76286 

.65973 

.75146 

.67230 

.73933 

.68561 

.72797 

.69321 

.71590 

43 

13 

.64679 

.76267 

.66000 

.75126 

.67301 

.73963 

.68532 

.72777 

.69342 

.71569 

42 

19 

.64701 

.76248 

.66022 

.75107 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229 

.66044 

.75083 

.67344 

.73924 

.68624 

.72737 

.69883 

.71.529 

40 

21 

.64746 

.76210 

.66066 

.75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71503 

39 

22 

.64763 

.76192 

.66088 

.75050 

.67387 

.73835 

.68666 

.72697 

.69925 

.71483 

38 

23 

.64790 

.76173 

.66109 

.75030 

.67409 

.73865 

.68683 

.72677 

.69946 

.71463 

37 

24 

.64812 

.76154 

.66131 

.75011 

.67430 

.73346 

.68709 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.66153 

.74992 

.67452 

.73826 

.68730 

.72637 

.69937 

.71427 

35 

26 

.643.56 

.76116 

.66175 

.74973 

.67473 

.73806 

.68751 

.72617 

.70003 

.71407 

34 

27 

.64873 

.76097 

.66197 

.74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

23 

.64901 

.76073 

.66213 

.74934 

.67516 

•73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

.76059 

.66240 

.74915 

.675.33 

.73747 

.63814 

.72557 

.70070 

.71345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73723 

.63835 

.72537 

.70091 

.71325 

30 

31 

.64967 

.76022 

.66234 

.74876 

.67530 

.73703 

.63357 

.72517 

.70112 

.71305 

29 

32 

.64939 

.76003 

.66306 

.74857 

.67602 

.73683 

.63373 

.72497 

.70132 

.71234 

23 

33 

.6.5011 

.75984 

.66-327 

.74838 

.67623 

.7.3669 

.63399 

.72477 

.70153 

.71264 

27 

34 

.6.5033 

.75965 

.66349 

.74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

.75946 

.66371 

.74799 

.67666 

.7.3629 

.68941 

.72437 

.70195 

.71223 

25 

36 

.6.5077 

.75927 

.66393 

.74730 

.67633 

.73610 

.63962 

.72417 

.70215 

.71203 

24 

37 

.65100 

.75908 

.66414 

.74760 

.67709 

.7.3.590 

.63933 

.72397 

.702.36 

.71182 

23 

33 

.65122 

.75389 

.66436 

.74741 

.67730 

.73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

.75370 

.66458 

.74722 

.67752 

.73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

.75851 

.66480 

.74703 

.67773 

.73531 

.69046 

.723.37 

.70293 

.71121 

20 

41 

.65133 

.75832 

.66501 

.74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

.75813 

.66523 

.74664 

.67316 

.73491 

.69033 

.72297 

.703.39 

.71030 

18 

43 

.65232 

.75794 

.66545 

.74644 

.67337 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.652.54 

.75775 

.66566 

.74625 

.67359 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

.75756 

.66533 

.74606 

.67330 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65298 

.75733 

.66610 

.74536 

.67901 

.73413 

.69172 

.72216 

.70422 

.70993 

14 

47 

.65320 

.75719 

.66632 

.74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65.342 

.75700 

.66653 

.74.543 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

.75680 

.66675 

.74523 

.67965 

.73353 

.69235 

.72156 

.70434 

.70937 

11 

50 

.6.5336 

.7.5661 

.66697 

.74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65403 

.75642 

.66718 

.74439 

.68008 

.73314 

.69277 

.72116 

.70525 

.70396 

9 

52 

.65430 

.75623 

.66740 

.74470 

.63029 

.73294 

.69293 

.72095 

.70546 

.70875 

8 

53 

.6.5452 

.7.5604 

.66762 

.74451 

.68051 

.73274 

.69319 

.72075 

.70567 

.70355 

7 

54 

.65474 

.75535 

.66733 

.74431 

.68072 

.73254 

.69340 

.72055 

.70537 

.703.34 

6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70603 

.70313 

5 

56 

.65518 

.75547 

.66327 

.74392 

.63115 

.73215 

.69382 

.72015 

.70623 

.70793 

4 

57 

.65540 

.75528 

.66343 

.74373 

.63136 

.73195 

.69403 

.71995 

.70649 

.70772 

3 

58 

.65562 

.75509 

.66370 

.74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65534 

.75490 

.66891 

.74334 

.63179 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 
M. 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.719^4 

.70711 

.70711 

0 

Cosin. 

Sine. 

Cosin.  Sine. 

Cosin.  1 

Sine.  Cosin.  1 

Sine. 

Cosin. 

Sine. 

493    1 

4:83           473      1      4:63     | 

4:53     1 

Tvnr^^ 


TABLE    XV. 


NATURAL   TANGENTS   AND    COTANGENTS 


230       TABLE   XV.       NATURAL  TANGENTS  AMU  COTANGENTS. 


M. 

0 

03             1 

1 

.0 

ao             1 

30 

M. 

60 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 
23.6363 

Tang. 

Cotang. 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

.0.5241 

19.0811 

1 

.00029 

3437.75 

.01775  1 

56.3506 

.03521 

23.3994 

.05270 

18.9755 

59 

2 

.00053 

1713.57 

.01304  i 

5.5.4415 

.03550 

28.1664 

.'05299 

18.8711 

58 

3 

.00087 

1145.92 

.01333 

54.. 56 13 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01362 

53.7036 

.03609 

27.7117 

.05357 

18.66.56 

56 

5 

.00145 

637.549 

.01391 

52.8321 

.03633 

27.4899 

05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0307 

.03667 

27.2715 

/J5il6 

.•e.4645 

54 

7 

.00204 

491.106 

.01949  i 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.713 

.01978 

50.5485 

.03725 

26.5450 

.05474 

18.2677 

52 

9 

.00262 

33L971 

.02007 

49.3157 

.03754 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

13.0750 

50 

11 

.00320 

312.521 

.(12066 

43.4121 

.03312 

26.2296 

.05562 

17.9502 

49 

12 

.00349 

2-;6.473 

.02095 

47.7395 

.0.3342 

26.0.307 

0.5591 

17.5563 

48 

13 

.00373 

264.441 

.02124 

47.0353 

.03371 

25.3348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02L53 

46.4439 

.03900 

25.6413 

.05649 

17.7015 

46 

15 

.00436 

229.132 

.02132 

45.3294 

.03929 

2.5.4517 

.05678 

17.6106 

45 

16 

.00465 

214.858 

.O23I0 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

44.6336 

.03937 

25.0798 

.05737 

17.4314 

43 

13 

.00524 

190.934 

.02269 

44.0661 

.04016 

24.8973 

.05766 

17.3432 

42 

19 

.00553 

130.932 

.02298 

43.5031 

.04046 

24.7135 

.05795 

17.2553 

41 

20 

.00532 

171. 3S5 

.02323 

42.9641 

.04075 

24.5413 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.0.5354 

17.0537 

39 

22 

.00640 

]  56.259 

02336 

41.9153 

.04133 

24.1957 

.05833 

16.9990 

38 

23 

.00669 

149.465 

.02415 

43.4106 

.01162 

24.0263 

.0.5912 

16.9150 

37 

24 

.00693 

143.23? 

.f'2444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

1^7.507 
152.219 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

.02502 

39.9655 

.042.50 

2.3.5.321 

.05999 

16.6631 

34 

27 

.00735 

127.  .321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5374 

33 

23 

.00315 

122.774 

.02560 

39.0563 

.04.303 

2.3.2137 

.06053 

16.5075 

32 

29 

.00344 

118.510 

.02589 

33.6177 

.043.37 

23.0577 

.06037 

16.4233 

31 

30 

.00S73 

114.5S9 

.02619 

38.1885 

.04366 

22.90.33 

.06116 

I6..3499 

30 

31 

.00902 

110392 

.02643 

37.7636 

.04395 

22.7519 

.06145 

16.2722 

2J 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6920 

.06175 

16.19.52 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00939 

101.107 

.02735 

36.5627 

.04483 

22.3031 

.06233 

16.04-35 

26 

35 

.01013 

93.2179 

.02764 

36.1776 

.04512 

■22.1fr40 

.06262 

15.9637 

25 

36 

.01047 

95.4395 

.02793 

35.3006 

.04541 

22.0217 

.06291 

15.3945 

24 

37 

.01076 

92.90S5 

.02322 

35.4313 

.04570 

21.3813 

.06321 

15.5211 

23 

33 

.01 lOo 

90.4633 

.02351 

35.0695 

.04599 

21.7426 

.06350 

15.7433 

22 

39 

.01135 

83.14.36 

.02881 

^4.7151 

.04623 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

35.9393 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6043 

20 

41 

.01193 

83.3435 

.02939 

^4.0273 

.04637 

21.3369 

.06437 

15.5-340 

19 

42 

.01222 

31.3470 

.02963 

33.6935 

.04716 

21.2049 

.06467 

15.46.33 

18 

43 

.01251 

79.94:34 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01230 

73.1263 

.03026 

33.0452 

04774 

20.9460 

.06.525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04303 

2^3183 

.06554 

15  2571 

15 

46 

.01333 

74.7292 

.03034 

32.4213 

.04333 

20.6932 

.06.584 

1-5.1593 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

205691 

.06613 

15.1222 

13 

48 

01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5234 

.04920 

20..3253 

.06671 

14.9398 

11 

50 

.01455 

63.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01434 

67.4019 

.0.32.30 

309599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.032.59 

30.6333 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.3.530 

.03233 

30.4116 

.05037 

19.3546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.0-3317 

30.1446 

.05066 

19.7403 

.06317 

14.6655 

6 

55 

.01600 

62.4992 

.03346 

29.3323 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

■  61.3329 

.0.3376 

29.6245 

.O0I24 

!    19.5156 

.06376 

14.54-33 

4 

57 

.016.58 

60.3053 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4523 

3 

53 

.01637 

59.2659 

.03434 

29.1220 

.05182 

19.29.59 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.0.3463 

23.3771 

.05212 

19.1579 

.06963 

14.3607 

1 

60 

m: 

.01746 
Co  tang. 

57.29flCi 

.03492 

23.6363 
Tang. 

.05241 

19.0311 

.06993 

14.. 3007 

0 
M. 

Tang. 

Cotang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

i 

93 

8 

§3 

g 

yo 

g 

60 

TABLt 

,   XV. 

NATURAL  TANGENTS 

AND  COTANGENTS. 

231 

M 

0 

4rO 

50 

60 

70 

M. 

60 

.    TaDg 

.06993 

1  Cotang 
14.3au7 

Tang. 

Cotang. 

Taug. 

Cotang. 
9.51436 

Tang. 
.12273 

Cotang. 
8.14435 

.03749 

11.4301 

.10510 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.4^731 

.  1 2.303 

8.12431 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12333 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08366 

11.27.39 

.10623 

9.40904 

.12397 

8.06674 

56 

5 

,  .07139 

14.0079 

.03895 

11.2417 

.10657 

9.3^307 

.12426 

8.04756 

55 

6 

J  .07168 

13.9.507 

.03925 

11.2  t4> 

.10637 

9.35724 

.12456 

8.02,348 

54 

7 

.07197 

13.8940 

.0j954 

11.1631 

.10716 

9.33155 

.12435 

8.00948 

"  ^       1 

53 

8 

•    .07227 

13.8378 

.03933 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

S 

.072.56 

1.3.7821 

.09013 

11.09.54 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.072S5 

13.7267 

.09042 

!  1.0594 

.10305 

9.25530 

.12574 

7.95302 

60 

11 

.07314 

13.6719 

.09071 

11.02.37 

.10:^34 

9.23016 

.12603 

7.93433 

49 

12 

i  .07314 

13.6174 

.09101 

10.93^2 

.10j63 

9.20516 

.12633 

7.91582 

48 

13 

1  .07373 

13.5634 

.091.30 

10.9529 

.10>93 

9.13028 

.12662 

7.89734 

47 

14 

i   .07402 

13.5093 

.091.59 

10.9178 

.10922 

9.15554 

.12692 

7.37895 

46 

15 

1   .07431 

13.4566 

.09189 

10.8329 

10952 

9.13093 

.12722 

7.S6C64 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10616 

.12751 

7.84242 

44 

17 

.07490 

1.3.. 351 5 

.09247 

10.8139 

.11011 

9.03211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.74.57 

.11070 

9.03379 

.12840 

7.78325 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00953 

.12369 

7.770.35 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.93598 

.12-99 

7.75254 

39 

22 

.076:^6 

13.0953 

.09394 

10.64.50 

.11158 

8.S6227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.9.3367 

.129.58 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.57^9 

.11217 

8.91520 

.12938 

7.69957 

36 

25 

.07724 

12.9469 

.09432 

10.. 5462 

.11246 

•8.39135 

.13017 

7.63208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.36362 

.13047 

7.66466 

1 

34 

27 

.07782 

12.3496 

.09541 

10.4313 

.11305 

8.34551 

.13076 

7.647.32 

33 

23 

.07812 

12.8014 

•09570 

10.4491 

.11335 

8.82252 

.13106 

7.6.3005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.131.36 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3354 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07^99 

12.6.591 

.09658 

10.  .3533 

11423 

3.7.5425 

.13195 

7.57372 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.. 56  J  76 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.63701 

.1.3234 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

.  10.2294 

.11.541 

8.66432 

.1.3313 

7.51132 

25 

36 

.08046 

12.4233 

.09305 

10.1938 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.03075 

12.3333 

.09334 

10.1683 

.11600 

8.62078 

.1.3372 

7.47306 

23 

33 

.03104 

12.. 3.390 

.09364 

10.1381 

.11629 

8.. 59893 

.13402 

7.46154 

22 

39 

.081.34 

12.2946 

.09893 

10.1080 

.116.59 

8.57718 

.134.32 

7.44509 

21 

40 

.03163 

12.2505 

.09923 

10.0780 

.11633 

8.55555 

.1.3461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0433 

.11718 

8.53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09931 

10.0137 

.11747 

8.512.59 

.13521 

7.39616 

18 

f. 

.0823 I 

12.1201 

.10011 

9.93931 

.11777 

8.49128 

.13550 

7.37999 

17 

44  1 

.08230 

12.0772 

.10040 

9.96G07 

11806 

8.47007 

.13.580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44396 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11365 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10123 

9.87333 

.11395 

8.40705 

.13669 

7.31600 

13 

48 

.03397 

11.9037 

.10153 

9.S44S2 

.11924 

S..3362.5 

.13693 

7.30013 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.23442 

11 

50 

.03456 

11.8262 

.10216 

9.78817 

.11933 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7353 

.10246 

9.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12012 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9,70441 

.12072 

8.2-3376 

.13346 

7.22204 

7 

54 

.03573 

11.6645 

.10.334 

9.67680 

.12101 

8.26355 

.13376 

7.20661 

6 

55 

.08602 

11.6243 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5353 

.10393 

9.62205 

.12160 

3.22.344 

.139,35 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20.352 

.1.3965 

7.18071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

13995 

7.14553 

2 

59 

.03720 

11.4635 

.10481 

9.54106 

.12249 

8.16393 

.  14024 

7.13042 

1 

6ii     .(Lsz-jy 

11  4301 

.10510     9.51436 

.12278 

8.144.35 
Tang.    ( 

.14054 

7.11.537 

0 

1  i 

M.  Cotang. 

Tang.     ( 

Jotang.  1    Tang.    ( 

[Jotang.  J 

Cotang. 

Tang.      1 

^_-. 

w.: 

i^ 

840 

833            1             8JJ0 

'46:< 

;    TAP 

!LE  XV. 

I^JATURAL  TANGENTS  AND  COTANGENTS 

). 

M 

0 

80 

9^ 

lOO 

110 

1 
M. 

60 

Tang. 
.14054 

CotaDg. 
7.11537 

Tang. 

Cotang. 

Tang. 

Cotang. 

5.67128 

Tang. 

Cotang. 
5.144.55 

.15333 

6.31375 

.176.33 

.19438 

1 

.14084 

7.10038 

15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.03546 

.15398 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

6.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5,11279 

56 

5 

.14202 

7.04105 

.15988 

6.25436 

.17783 

5.62344 

.19539 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17343 

5.60452 

.19649 

5.03921 

63 

8 

.14291 

6.99713 

.16077 

6.22003 

.17373 

5.59511 

.1L630 

5.031.39 

52 

9 

.14321 

6.9326S 

.16107 

6.20351 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96323 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.143S1 

6.95335 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.0.5809 

49 

12 

.14410 

6.9.3952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16233 

.13023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.39683 

.16236 

6.14023 

.18083 

5.53007 

.19391 

5.02734 

46 

16 

.14529 

6.88278 

.16316 

6.12399 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86374 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.13173 

5.50264 

.19952 

5.00451 

42 

19 

.14618 

6.84032 

.16405 

6.09.552 

.13203 

5.49356 

.20012 

4.99695 

41 

20 

.14643 

6.82694 

.16435 

6.03444 

.13233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.SI3I2 

.16465 

6.07340 

.13263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46643 

.20103 

4.97433 

38 

23 

.14737 

6.73564 

.16525 

6.05143 

.13323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04J51 

.13353 

5.44357 

.20164 

4.95945 

36 

25 

.14796 

6.75S33 

.16535 

6.02962 

.18334 

5.43966 

.20194 

4.9.5201 

35 

26 

.14326 

6.74433 

.16615 

6.01 37S 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14S56 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

23 

.14336 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92934 

32 

29 

.14915 

6.70450 

.16704 

5.93646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67737 

.16764 

5.96510 

.18564 

5.33677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.9.S448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16324 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.1.5064 

6.6.3331 

.16354 

5.9a335 

.18654 

5.36070 

.20466 

4.SS605 

26 

35 

. 1 5094 

6.62523 

.16381 

5.92283 

.18634 

5.35206 

.20497 

4.87882 

26 

36 

.15124 

6.61219 

16914 

5.91236 

.18714 

5.»4345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33437 

.20557 

4.86444 

23 

33 

.15183 

6.53627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87030 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5. 8605 1 

.18865 

5.30030 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82382 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.23393 

.20739 

4.82175 

17 

44 

.15362 

6.. 50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18936 

5.26715 

.20300 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80063 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24213 

.20891 

4.7S673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43434 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77236 

10 

51 

.15570 

6.42253 

.17363 

5 

75941 

.19166 

.5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5 

74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5 

73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.1.5660 

6.33587 

.17453 

5 

72974 

.192.57 

5.19293 

.21073 

4.74534 

6 

55 

.15639 

6.37374 

.17483 

5.71992 

.19237 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

67 

.15749 

6.34961 

.17543 

5.70037 

.19:347 

5.16363 

.21164 

4.72490 

3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16053 

.21195 

4.71813 

2 

59 

.15309 

6.32566 

.17603 

5.63094 

.19403 

5.15256 

.21225 

4.71137 

1 

60 
1 

.15838 

6.31375 

.17633 
Cotang. 

5.67123 

.19433 

5.14455 

.21256 

4.70463 

0 
ftl. 



Cotang. 

Tang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

81°            1 

803            1 

793            1 

783            1 

TABLE  XV.      NATURAL  TANGENTS  AND  COTANGENTS.       233 


M. 

0 

130 

130     1 

1*0 

150 

M. 

60 

Tang. 

Cotang. 

Tang. 

Cotang.  ^ 

Cang. 

Cotaug. 

Tang. 

Cotang. 

21256 

4.70463 

.23087 

4.33148  . 

24933 

4.01078 

.26795 

3.73205 

1 

.2I2S6 

4.69791 

.23117 

4.32573  . 

24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001  . 

24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.314.30  . 

25026 

3.99592 

.26883 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30S60  . 

25056 

3.99099 

.26920 

3.71476 

66 

5 

.21408 

4.67121 

.2.3240 

4.30291  . 

25087 

3.93607 

.26951 

3.71046 

55 

6 

.21433 

4.66458 

.23271 

4.29724  . 

25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.2.3301 

4.29159  . 

25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65133 

.23332 

4.2S595  . 

25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032  . 

25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4  63825 

.23393 

4.27471  . 

25242 

3.96165 

.27107 

3.68909 

60 

11 

.21.590 

4  63171 

.23424 

4.26911  . 

25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352  . 

25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4. 6 1 868 

.23485 

4.25795  . 

253.35 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.252.39  . 

25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685  . 

25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132  . 

25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.. 592-^3 

.23608 

4.23580  . 

2.5459 

3.92793 

.27326 

3.65957 

43 

IS 

.21804 

4.58641 

.23639 

4.23030  . 

25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.55001 

.23670 

4.22481  . 

25521 

3.91839 

.27388 

3.65121 

41 

2() 

.21864 

4.57363 

.23700 

4.21933  . 

25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387  . 

25583 

3.90890 

.27451 

3.64289 

39 

22 

.21925 

4.56091 

.23762 

4.20842  . 

2.5614 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20293  . 

25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.. 54826 

.23823 

4.19756  . 

25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215  . 

25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.23885 

4.18675  . 

25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.2.3916 

4.18137  . 

25769 

3.88068 

.27633 

3.61814 

33 

28 

.22108 

4..52316 

.23946 

4.17600  . 

25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064  . 

2.5831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530  . 

25862 

3.86671 

.27732 

3.605SS 

30 

31 

.22200 

4.50451 

.24039 

4.15997  . 

25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465  . 

25924 

3.85745 

.27795 

3..59775 

28 

33 

.22261 

4.49215 

.24100 

4.149.34  . 

25955 

3.85284 

.27826 

3.59370 

27 

34 

22292 

4.48600 

.24131 

4.14405  . 

259S6 

3.84824 

.27858 

3.58966 

26 

35 

22322 

4.47986 

.24162 

4.13877  . 

26017 

3.84.364 

.27889 

3.55562 

25 

36 

22.353 

4.47374 

.24193 

4.13350  . 

26048 

3.83906 

.27921 

3.58160 

24 

37 

22383 

4.46764 

.24223 

4.12825  . 

26079 

3.8.3449 

.27952 

3.57758 

23 

38 

22414 

4.46155 

.24254 

4.12301  . 

26110 

3.82992 

.27933 

3.-57357 

22 

39 

22444 

4.45548 

.24285 

4.11778  . 

26141 

3.82.537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256  . 

26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44.3.38 

.24347 

4.10736  . 

26203 

3.81630 

.28077 

3.561.59 

19 

42 

.22536 

4.4.3735 

.24377 

4.10216  . 

26235 

3.81177 

.28109 

3.55761 

18 

43 

.22567 

4.43134 

.21408 

4.09699  . 

26266 

3.80726 

.28140 

3.  .55364 

17 

44 

.22597 

4.42534 

.244.39 

4.09182  . 

26297 

3.80276 

.28172 

3.54963 

16 

45 

.22623 

4.41936 

.24470 

4.0S666  . 

26328 

3.79S27 

.28203 

3.  .54573 

15 

46 

.22658 

4.41.340 

.24.501 

4.08152  . 

26359 

3.79.378 

.23234 

3..54179 

14 

47 

.22689 

4.40745 

.24.532 

4.076.39  . 

26390 

3.78931 

.28266 

3.5.3785 

13 

48 

.22719 

4.401.52 

.24562 

4.07127  . 

26421 

3.78485 

.28297 

3.. 53393 

12 

49 

.22750 

4.39560 

.24.593 

4.06616  . 

26452 

3.78040 

.2S329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107  . 

26483 

3.77.595 

.2*360 

3.52609 

10 

51 

.22811 

4.3S381 

.246.55 

4.05599  . 

26515 

3.771.52 

.28.391 

3.  .522 19 

9 

52 

.22842 

4.37793 

.246S6 

4.05092  . 

26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586  . 

26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4..36623 

.24747 

4.04081  . 

26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578  . 

26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.354.59 

.24809 

4.03076  . 

26670 

3.749.50 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574  . 

26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074  . 

26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.. 33723 

.24902 

4.01576  . 

26764 

3.73640 

.28643 

3.49125 

1 

60 

m: 

.23(:87 

4..3:3143 

.24933 

4.01078  . 

26795 

3.73205 

.28675 

3.48741 

0 
M. 

Co tang. 

Tang. 

Cotang. 

Tang.  C 

:tang. 

Tang. 

Cotang. 

Tang. 

i 

TO 

reo   1 

750 

7 

4:0 

u;j4 

.       TABLE   XV. 

NATl 

URAL  TANGENTS  AND 

COTANGENTS 

• 

M. 

0 

160 

170 

18^ 

190 

M. 

60 

Tang. 

.23675 

Cotang. 

Tang. 

Cotang. 

Tang. 

.32492 

I  Cotang. 
3.07763 

Tang. 

Cotang. 

2.90421 

3.43741 

.30573 

3.27035 

-34433 

1 

.28706 

3.43.359 

.30605 

3.26745 

.32.524 

3.07464 

.34465 

2.90147 

59 

2 

.23738 

3.47977 

.-30637 

3.26406 

..32556 

3.07160 

.34493 

2.89373 

58 

3 

.23769 

3.47596 

.30669 

3.26067 

.32533 

.3.06357 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.23832 

3.46337 

.-307-32 

3.25392 

-32653 

3.062-52 

-34596 

2.89055 

55 

6 

.23S&4 

3.46453 

..30764 

3.25055 

.32635 

3.059-50 

.34628 

2.83783 

54 

7 

.23S95 

3.46030 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.8351 1 

53 

8 

.23927 

3.45703 

.30823 

3.24333 

.32749 

3.05349 

.34693 

2.33240 

52 

9 

,  .28953 

3.45.327 

.30360 

3.24049 

.32732 

3.05049 

.-34726 

2.87970 

51 

10 

.28990 

3.44951 

.-30391 

3.2.3714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23-331 

.32346 

3.04450 

.34791 

2.S7430 

49 

12 

.29053 

3.44202 

.30955 

3.2304S 

.32373 

-3.041.52 

.34824 

2.37161 

43 

13 

.29034 

3.43323 

.3'i9';7 

3.22715 

.32911 

3.0.3354 

.34356 

2.86392 

47 

14 

.29116 

3.434.56 

.31019 

3.22-334 

.32943 

3.03556 

-343S9 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

..32975 

3.03260 

.34922 

2.S6356 

45 

16 

.29179 

3.42713 

.31083 

.3.21722 

.a3007 

3.02963 

.349.54 

2.86039 

44 

17 

.29210 

3.42343 

.31115 

,3.21-392 

.33040 

3.02667 

.34987 

2.85822 

43 

13 

.29242 

.3.41973 

.31147 

.3.21063 

.33f)72 

3.02372 

.3-5020 

2.85555 

42   : 

19 

.29274 

.3.41604 

.31178 

3.20734 

-33104 

3.02077 

..35052 

2.8.5239 

41 

20 

.29305 

3.412.36 

.31210 

3.20406 

.331-36 

3.01733 

.3.5035 

2.85023 

40    1 

21 

.29337 

3.40^69 

.31242 

3.20079 

.-33169 

3.01439 

.35113 

2.S475S 

39  ; 

22 

.29363 

3.40502 

.31274 

-3.19752 

..33201 

.3.01196 

.351.50 

2.84491 

38 

23 

.29400 

3.401.36 

.31306 

-3-19426 

.3.3233 

3.00903 

.35133 

2.8-1229 

37    ! 

24 

.29432 

3.. 39771 

.31333 

3.19100 

.-33266 

3.00611 

.3.5216 

2.83965 

36    i 

25 

.29463 

3.. 39406 

.31370 

3.13775 

.33293 

3.00319 

.3.5248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.13451 

.33330 

3.00023 

.35231 

2.83439 

34 

27 

.29.526 

3.33679 

.314-^ 

3  18127. 

.-33.363 

2.997.33 

.35314 

2.83176 

33 

23 

.29558 

3..38317 

.31466 

-3.  i  7304 

.33-395 

2.99447 

.3.5346 

2.82914 

32 

29 

.29590 

3.37955 

-31493 

3.17431 

.-3-3427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.93363 

..3-5412 

2.82391 

.30 

31 

.29653 

3.-372.34 

.31562 

3.16333 

.3-3492 

2.93530 

.35445 

2.82130 

29 

32 

.29635 

3.-36375 

.31.594 

3.16517 

.33524 

2.93292 

.35477 

2.81870 

23 

33 

.29716 

3.36516 

.31626 

3.16197 

.335-57 

2.93004 

.3.5510 

2.SI610 

27 

34 

.29743 

3.36153 

.31653 

3.15377 

.3-3559 

2.97717 

.-35-543 

2.81350 

26 

35 

.29730 

3.35800 

.31690 

3.15558 

-33621 

2.974-30 

.35576 

2.81091 

25 

36 

.29311 

3.35443 

.31722 

3.15240 

.-3-36.54 

2.97144 

..35603 

2.60333 

24 

37 

.29343 

3.  .3-5037 

.31754 

3.14922 

.33636 

2.963-58 

.3.5841 

2.S0574 

23 

3S 

.29875 

3.347-32 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.-33751 

2.96283 

..35707 

2.80059 

21 

40 

.29933 

3.ai023 

.31850 

3.1-3972 

.-33733 

2.96004 

..35740 

2.79S02 

20 

41 

.29970 

3.33670 

.31832 

3.136-56 

..3-3316 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.-3-3317 

.31914 

3.13-341 

.3-3348 

2.9-5437 

..35305 

2.79289 

18 

43 

..30033 

3.32965 

.31946 

3.1.3027 

..3-3381 

2.951-55 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31973 

3.12713 

-33913 

2.94372 

.35371 

2.78773 

16 

45 

..30097 

3.-32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30128 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.30160 

3.31-565 

.32074 

3.11775 

.34010 

2.94028 

.35969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.9-3748 

.36002 

2.77761 

12 

49 

.30224 

3. -3036  3 

.32139 

3.11153 

.34075 

2.93463 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10342 

.34108 

2.93189 

.36068 

2.772.54 

10 

51 

.30237 

3.-30174 

..32203 

3.10532 

.34140 

2.92910 

..36101 

2.77002 

9 

52 

.30319 

3.29329 

..32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30.351 

3.29433 

..32267 

3.09914 

.31205 

2.92.3-54 

.36167 

2.76493 

7 

54 

.30382 

-3.291-39 

.32299 

3.09606 

.34233 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.23795 

.-32-331 

3.09295 

.34270 

2.91799 

.362-32 

2.75996 

5 

56 

.30446 

3.234-52 

.32.363 

3.03991 

.34303 

2  91.523 

..36265 

2.75746 

4 

57 

.30478 

3.23109 

.32396 

3.03635 

.343.35 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08-379 

..34363 

2.90971 

.36331 

2.75246 

2 

59 

.30.541 

3.27426 

.32460 

3.03073 

.34400 

2.90696 

.36364 

2.74997 

1 

60 
M. 

.30573 

3.27035 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74743 

0 
M. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

T33            1 

73°           1 

71^            \ 

703            1 

TABLE  X\^ 


NATURAL  TANGENTS  AND  COTA.'JGENTS.   235 


M. 


20^ 


31^ 


/ 

8 

9 
10 
11 
12 
13 
14 
15 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
23 
29 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
53 
59 
60 

m; 


T36397 
.36430 
.36463 
.364% 
.36529 
.36562 
.36595 
.36628 
.36661 
.36694 
.36727 
.36760 
.36793 
.36S26 
.36859 
.36892 

.36925 
.36958 
.36991 
.37024 
.37057 
.37090 
.37123 
.37157 
.37190 
.37223 
.37256 
.37289 
.37322 
.37355 
.37333 

.37422 
.37455 
.37438 
.37521 
.37554 
.37583 
.37621 
.37654 
.37687 
.37720 
.37754 
.37787 
.37820 
.37353 
.37837 

.37920 
37953 
.37936 
.33020 
.38053 
.33036 
33120 
.33153 
.33186 
.  3^221) 
.3=!2.-)3 
.3-2-6 
.3-320 
.3-353 
.38:386 

Cotang. 


Cotang.  Tang. 


2.74748 
2.74499 
2.74251 
2.74004 
2.73756 
2.73509 
2.73263 
2.73017 
2.72771 
2.72526 
2.72281 
2.72036 
2.71792 
2.71548 
2.71305 
2.71062 

2.70819 
2.70577 
2.70335 
2.70094 
2.69853 
2.69612 
2.69371 
2.69131 
2.68892 
2.68653 
2.63414 
2.63175 
2.67937 
2.67700 
2.67462 

2.67225 
2.66939 
2.66752 
2.66516 
2.66231 
2.66046 
2.65811 
2.6.5576 
2.65342 
2.65109 
2.64875 
2.64642 
2.64410 
2.64177 
2.63945 

2.63714 

2.634S3 

2.63252 

2.63021 

2.62791 

2.62.561 

2  62332 

2.62103 

2.61^74 

2.61646 

2.61418 

2.61190 

2  60963 

2.60736 

2.60509 


Cotang. 


.33336 
.33420 
.33453 
.38487 
.38520 
.38553 
.38587 
.38620 
.33654 
.33687 
.3^721 
.33754 
.38787 
.38321 
.33354 
.33888 

.33921 
.33955 
.38988 
..39022 
.390.55 
.39089 
.39122 
.39156 
.39190 
.39223 
.39257 
.39290 
.39324 
.39357 
.39391 

.39425 
.39453 
.39192 
.39526 
.39559 
.39593 
..39626 
.39660 
.39694 
.39727 
.39761 
.39795 
.39829 
.39362 
.39896 

..39930 
.39963 
.39997 
.40031 
.40065 
.40093 
.401,32 
.40166 
.40200 
.40234 
.40267 
.40301 
.40335 
.40369 
.40103 


a^j 


23C 


Tang. 


Tang. 


69= 


2.60509 
2.60283 
2.60057 
2.59331 
2.59606 
2.593>1 
2.59156 
2.53932 
2.58703 
2.53434 
2.53261 
2.58038 
2.57815 
2.57593 
2.57371 
2.57150 

2.56923 
2.56707 
2.56487 
2.56266 
2.56046 
2.55827 
2.55608 
2.553S9 
2.55170 
2.54952 
2.54734 
2.54516 
2.54299 
2.b40S2 
2.53365 

2.53643 
2.-53432 
2.53217 
2.53001 
2.52786 
2.52571 
2.52357 
2.52142 
2.51929 
2.51715 
2.51502 
2.51289 
2.51076 
2.50364 
2.50652 

2.50440 
2.50229 
2  50018 
2.49807 
2.49.597 
2.49336 
2.49177 
2.4-!967 
2.487.58 
2.43549 
2.48340 
2.48132 
2.47921 
2.47716 
2.47509 


Cotang.  Tang. 
68= 


.40403 
.40136 
.40470 
.41)504 
.40538 
.40572 
.40606 
.40640 
.40674 
.40707 
.40741 
.40775 
.40309 
.40843 
.40377 
.4091 1 

.40945 
.40979 
.41013 
.41047 
.41081 
.41115 
.41149 
.41183 
.41217 
.41251 
.41235 
.41319 
.41353 
.41337 
.41421 

.41455 
.41490 
.41524 
.41558 
.41592 
.41626 
.41660 
.41694 
.41723 
.41763 
.41797 
.41331 
.41365 
.41399 
.41933 

.41963 
.42002 
.420.36 
.42070 
.42105 
.42139 
.42173 
.42207 
.42242 
.42276 
.42310 
.42345 
.42379 
.42413 
.42447 


Cotang. 

■2.47509 
2.47302 
2.47095 
2.46883 
2.46632 
2.46476 
2.46270 
2.46065 
2.45860 
2.45655 
2.45451 
2.45246 
2.45043 
2.44^39 
2.44636 
2.44433 

2.44230 
2.44027 
2.43325 
2.43623 
2.43422 
2.43220 
2.43019 
2.42819 
2.42618 
2.42418 
2.42213 
2.42019 
2.41819 
2.41620 
2.41421 

2.41223 
2.41025 
2.40827 
2.40629 
2.40432 
2.40235 
2.40033 
2.39841 
2.39645 
2.39449 
2.392.53 
2.39053 
2.33>63 
2.3^66^ 
2.33473 

2.38279 
2.. 33034 
2.. 37891 
2.37697 
2.37504 
2.37311 
2.37118 
2.36925 
2.367.33 
2.36541 
2.36349 
2.36158 
2.35967 
2.35776 
2.35585 


Tang.  I  Cotang. 


.42147 
.424^2 
.42516 
.42.551 
.425S5 
.42619 
.42654 
.42638 
.42722 
.42757 
.42791 
.42826 
.42-:'60 
.42394 
.42929 
.42963 

.42998 
.43032 
.43067 
.43101 
.431.36 
.43170 
.43205 
.43239 
.43274 
.43308 
.43343 
.43378 
.43412 
.43447 
.43431 

.43516 
.43550 
.4.3535 
.43620 
.436.54 
.43639 
.43724 
.43758 
.43793 
.43328 
.43362 
.43897 
.43932 
.43966 
.44001 

.44036 
.44071 
.44105 
.44140 
.44175 
.44210 
.44244 
.44279 
.44314 
.44349 
.44334 
.44418 
.44453 
.444.83 
.44523 


Cotang.   Tang 
67= 


2.35585 
2.35395 
2.35205 
2.35015 
2.34825 
2.34636 
2.34447 
2..34258 
2.34069 
2.33881 
2  33693 
2  33505 
2,33317 
2.33130 
2.32943 
2.32756 

2..32570 
2.32383 
2.32197 
2.32012 
2.31826 
2.31641 
2.31456 
2.31271 
2. 31 086 
2.30902 
2.30718 
2.30.5.34 
2.30351 
2.30167 
2.29984 

2.29801 
2.29619 
2.29437 
2.29254 
2.29073 
2.28S91 
2.23710 
2.28523 
2.283-18 
2.28167 
2.27987 
2.27506 
2.27626 
2.27447 
2.27267 

2.27033 
2.26909 
2.26730 
2.265.52 
2.26374 
2.2615:6 
2.26013 
2.25840 
2.25663 
2.25436 
2.2.5309 
2.25132 
2.24956 
2.24780 
2.24604 


M. 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 

44 
43 
42 
41 
40 
39 
33 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cotang.     Tang 
663 


sat 

)       TABLE  XV. 

NATURAL  TANGENTS  AND  COTANGENTS 

!• 

M 

0 

1            340 

353 

2&0 

370 

1 

Tang. 
,  .44523 

Cotang. 

Taog. 

Cotang. 
2.14451 

Tang. 

.43773 

Cotang. 
2.05030 

Tang. 

Cotang. 

1.96261 

M. 

60 

2.24604 

.46631 

.50953 

1 

.44553 

2.24423 

.46666 

2. 142S3 

.43309 

2.04379 

.50939 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.43345 

2.04723 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.43Ssl 

2.04577 

.51063 

1.953.38 

57 

4 

.44662 

2.23902 

.46772 

2.13<01 

.43917 

2.04426 

.51099 

]. 95698 

56 

5 

.44697 

2.23727 

.46-03 

2.136-39 

.4^953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.2.3553 

.46>43'   2.1.3477 

43939 

2.04125 

.51173 

1.9.5417 

54 

7 

.44767 

2.233/-> 

.46379 

2.1.3316 

.49026 

2.0.3975 

.51209 

1.95277 

53 

8 

.44S02 

2.2321)4 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

■  .44337 

2.2.3030 

.46950 

2.12993 

.49093 

2.0.3675 

.51283 

1-94997 

51 

10 

.44372 

2.22S57 

.46935 

2.12332 

.491.34 

2.03526 

.51319 

1.94358 

50 

11 

.44907 

2.22633 

.47021 

2.12671 

.49170 

2.03376 

.51-356 

1.94718 

49 

12 

.44W2 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

43 

13 

'■  .44977 

2.22337 

.47092 

2.123-50 

.49242 

2.03073 

.514.30 

1.94440 

1  47 

14 

1  .45012 

2.22164 

.47123 

2.12190 

.49273 

2.02929 

.51467 

1.94.301 

46   , 

15 

:  .45047 

2.21992 

.47163 

2.12030 

.49315 

2.027S0 

.51503 

1.94162 

45 

16 

■  .450S2 

2.21319 

.47199 

2.1 1371 

.49351 

2.02631 

.51;540 

1.94023 

44 

17 

.45117 

2.21647 

.472-34 

2.11711 

.49337 

2.02433 

.51.577 

1.9.3885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.9.3746 

42 

19 

.45187 

2.21:304 

.47.305 

2.11.392 

.49459 

2.02187 

.51651 

1.9.3608 

41 

20 

.45222 

2.21132 

.47311 

2-11233 

.49495 

2.02039 

.51633 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01391 

.51724 

1.93-332 

39 

22 

.45292 

2.2)79:1 

.47412 

2.10916 

.49-563 

2.01743 

.51761 

1.93195 

38 

23 

.45.327 

2.20619 

.47443 

2.10753 

.49604 

2.01596 

.51798 

1.9.3057 

37 

24 

.4.5362 

2.20449 

.47433 

2.10600 

.49640 

2.01449 

.51835 

1.920-.40 

36 

25 

.45397 

2.20273 

.47519 

2.1W42 

.49677 

2.01302 

.51372 

1.92782 

35 

26 

.454.32 

2.20103 

.47555 

2.10234 

.49713 

2.01155 

.51909 

1  92&45 

34 

27 

.45467 

2.1993S 

.47590 

2.10126 

.49740 

2.01008 

.51946 

i.  92503 

33 

28 

.4.5502 

2.19769 

.47626 

2.09969 

.49736 

2.00=62 

.51983 

1.92371 

32 

29 

.45533 

2.19-599 

.47662 

2.09311 

.49322 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.194.30 

.47693 

2.09654 

.493.53 

2.00569 

..52057 

1.92093 

30 

31 

.45603 

2.19261 

.47733 

2.0349S 

.49394 

2.004V3 

.520G1 

1.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09.341 

.49931 

2.00277 

.52131 

1.91326 

28 

33 

.45673 

2.13923 

.47305 

2.09134 

.49967 

2.00131 

.55163 

1.91690 

27 

34 

.45713 

2.13755 

.47^^0 

2.09023 

.50004 

1.999-6 

.52205 

1.91554 

26 

35 

.45743 

2.13.537 

.47876 

2-03372 

..50040 

1.99341 

.52242 

1.91418 

25 

36 

.45731 

2.13419 

.47912 

2.03716 

..50076 

1.99695 

.52279 

1.91232 

24 

37 

.45319 

2.13251 

.47943 

2.03560 

.50113 

1.99.550 

..52316 

1.91147 

23 

38 

.453:54 

2.13034 

.47934 

2.03405 

..50149 

1.99406 

.52353 

1.91012 

22 

39 

.45389 

2.17916 

.43019 

2.03250 

.50135 

1.99261 

.52-390 

1.90376 

21 

40 

.4.5924 

2.17749 

.430.55 

2.03094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17.532 

.43091 

2.07939 

.50253 

1.93972 

.52461 

1.90607 

19 

42 

.45995 

2.17416 

.43127 

2.07785 

.50295 

1.93328 

.52.501 

1.90472 

18 

43 

.46030 

2.17249 

.43163 

2.076.30 

..50331 

I.986&} 

.525.38 

1.90.3.37 

17 

44 

.46065 

2.17033 

.43193 

2.07476 

.50363 

1.93540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.43234 

2.07321 

.50404 

1.93396 

.52613 

1.90069 

15 

46 

.461.36 

2.16751 

.43270 

2.07167 

.50441 

1.932.53 

.52650 

1.89935 

14 

47 

.46171 

2.16535 

.43306 

2-07014 

.50477 

1.93110 

.52637 

1.89301 

13 

4S 

.46206 

2.16420 

.43342 

2  06360 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.162.55 

.43378 

2.06706 

.50550 

1.97323 

..52761 

1.89.533 

11 

50 

.46277 

2.16090 

.43414 

2-06-5.53 

.50537  1 

1.97631 

.52793 

1 .89400 

10 

51  1 

.46312 

2.15925 

.43450 

2.06400 

.50623 

1.97.5.33 

.52836 

1.89266 

9 

52 

.46.343 

2.15760 

.43436 

2.06247 

.50660 

1  97395 

52373 

1.89133 

8 

53 

.46.333 

2.15.596 

.43521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.1.54.32 

.4S557 

2.05942 

.50733 

1.97111 

.52917 

1. 83367 

6 

55 

.464.54 

2.1.5263 

.43593 

2.05790 

.50769 

1.96969 

.52935 

1.83734 

5 

56 

.46439 

2.15104 

.43629 

2.05637 

.50806 

1.96327 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.43665 

2.05435 

.50843 

1.96635 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50379 

1.96544 

.53096 

1.83337 

2 

59 

.46595 

2.14614 

.43737 

2. 05 182 

..50916 

1.96402 

.53134 

1.88205 

1 

60; 

.46631 

2.14451 

.43773 

2.0.5030 
Tang.     ( 

.50953 
IJotang. 

1.96261 

..53171 

1.88073 
Tang.     ] 

0 

M.  Cotang. ! 

Tang. 

Cotang. 

Tang.    < 

;:!otang. 

i 

■s:: 

6i 

5C 

64°            1 

633            1 

633            1 

lABLE   XV.      NATURAL  TANGENTS  AND  COTA/JGENTS.       23T 


M 

0 

aso 

393 

30O 

310 

M. 

60 

Tang. 

.53171 

Cotang. 

Tang. 

Cotang. 

Tang. 
.57735 

Cotang. 
1.73205 

Tang. 

Cotang. 

1.88073 

.55431 

1.80405 

.60086 

1.6642-5 

1 

.53208 

1.87941 

.55469 

1.80231 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

1.87677 

.55.545 

1.80034 

.57851 

1.72357 

.60205 

1.66099 

57 

4 

.53320 

1.87546 

.55583 

1.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

1.87415 

.55621 

1.7978S 

.57929 

1.72625 

.602.34 

1.65S81 

55 

6 

.53395 

1.87233 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542 

..58007 

1.72393 

.60364 

1.65663 

53 

8 

..53470 

1.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86391 

.55774 

1.79296 

.53035 

1.72163 

.60443 

1.65-145 

51 

10 

.53545 

1.86760 

.5.5812 

1.79174 

.581^ 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55838 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78507 

..58240 

1.71702 

.60602 

1.6.5011 

47 

14 

.53694 

1.862.39 

.55964 

1.78635 

.58279 

1.71588 

.00642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

1 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53307 

1.85350 

.56079 

1.78319 

.53396 

1.71244 

.60761 

1.61579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

..58435 

1.71129 

.60801 

1.64471 

42 

19 

.5.3882 

1.85591 

.561.56 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

2n 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60381 

1.64256 

40 

21 

.53957 

1.85333 

..56232 

1.77^34 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204 

..56270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

33 

23 

.54032 

1.8.5075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

..56347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

25 

..54107 

1. 84318 

.56335 

1.77.351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.64683 

.53424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.51183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1. 63505 

33 

23 

.54220 

1.84433 

..56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.542:58 

1.84305 

.56539 

1.76369 

..58865 

1.69379 

.61240 

1.63292 

31 

80 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.696.53 

.61320 

1.63079 

29 

32 

.54.371 

1.83922 

.56651 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69423 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

..56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54434 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56303 

1.76032 

.59140 

1.69091 

.61.520 

1.62.548 

24 

37 

.54560 

1.83286 

.56346 

1.75913 

.59179 

1.63979 

.61561 

1.624-12 

23 

38 

.54597 

1.83159 

.56335 

1.75794 

.59218 

1.63866 

.61601 

1.62.336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.59258 

1.63754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.63643 

.61681 

1.62125 

20 

41 

.5^1711 

1.827S0 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1. 6201 9 

19 

42 

.54743 

1.82654 

.57039 

1.75319 

.59376 

1.63419 

.61761 

1.61914 

18 

43 

.54786 

1.82523 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61803 

17 

44 

.54324 

1.82402 

.57116 

1.7.5032 

.59154 

1.68196 

.618-12 

1.61703 

18 

45 

.54362 

1.82276 

.57155 

1.74964 

.59494 

1.63035 

.61882 

1.61593 

15 

46 

.54900 

1.82150 

.57193 

1.74346 

.59533 

1.67974 

.61922 

1.61493 

14 

47 

.549:« 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61338 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61233 

12 

49 

.5.5013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.01179 

11 

50 

.55051 

1.81649 

..57343 

1.71375 

.59691 

1.67530 

.62033 

1.61074 

10 

51 

.55a39 

1.81524 

.57336 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

6216^1 

1.60665 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.5.5203 

1.81150 

.57503 

1.7.3905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59883 

1.66978 

.62235 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

..59928 

1.66867 

.62.325 

1.60449 

4 

57 

.55317 

1.80777 

..57619 

1.7.3555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73138 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73.321 

.600-16 

1.66538 

.62446 

1.60137 

1 

60 
M. 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66423 

.62487 

1.60033 

0 
M. 

Cotang. 
6 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

602 

5 

.93 

5 

83 

238      TABLE  XV.      NATURAL  TANGENTS  AND  COTANGENTS. 


M 
0 

323 

33^            1 

34  ; 

3 

5^ 

M. 

60 

Tang. 
.624S7 

Cctang. 
1.600.33 

Tang. 

Cotang. 

Tang. 
.67451 

Cotang. 
1.45-2-56 

Tang. 

Cotang. 
1.4-2>15 

.64941 

1.5:39^6 

.70021 

1 

.62527 

I.. 59930 

.649-2 

1.53S5S 

.67493 

1.43 1 63 

.70C64 

1.42726 

59 

2 

.62.563 

1.59326 

.65024 

1.53791 

.675:36 

1.43070 

.70107 

1.426:33 

58 

3 

.62603 

1.59723 

.65065 

1.53693 

.67573 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.596-20 

.65106 

1.. 53595 

.67620 

1.47335 

.70194 

1.42462 

56 

5 

.62639 

1.59517 

.65143 

1.53497 

.67663 

1.47792 

.70233 

1.42374 

55 

6 

.62730 

1.59414 

.65159 

1.534G0 

.67705 

1.47699 

.70231 

1.42236 

54 

7 

.62770 

1.59311 

.65231 

1.5:3:302 

.67743 

1.47607 

.70325 

1.42198 

53 

8 

.62311 

1.59203 

.65272 

1.53205 

.67790- 

1.47514 

.70.363 

1.42110 

52 

9 

.62352 

1.59105 

.6.5314 

1.53107 

.67332 

1.47422 

.70412 

1.42022 

51. 

10 

.62392 

1.59002 

.65-3.55 

1.53010 

.67375 

1.47.3:30 

.70455 

1.419:3-1 

50' 

11 

.62933 

1.53900 

.6-5397 

1.52913 

.67917 

1.47233 

704S9 

1.41647 

49 

12 

.62973 

1.53797 

.65433 

1.52316 

.67960 

1.47146 

.705^42 

1.417.59 

48 

13 

.63014 

1.53695 

.65450 

1.52719 

.63002 

1.470.33 

.■70536 

1.41672 

47 

14 

.63055 

1.53.593 

.63521 

1.52622 

.63045 

1.46932 

.70629 

1.41-5.34 

46 

15 

.63095 

.1.53490 

.6-5563 

1.52525 

.63033 

1.46370 

.70673 

1.41497 

45 

16 

.6-3136 

1.5S333 

.65604 

1.52429 

.63130 

1.46773 

.70717 

1.41409 

44 

17 

.63177 

1.53236 

.65646 

1.523:32 

.63173 

1.46656 

.70760 

1.41322 

43 

13 

.63217 

1.53134 

.65633 

1.522-35 

.63215 

1.46595 

.70804 

1.41-235 

42 

13 

.63253 

1.53033 

.65729 

1. 52139 

.65-253 

1 .46503 

.70348 

1.41148 

41 

20 

.63299 

1.57931 

.65771 

1.52043 

.6530! 

1.46411 

.70391 

1.41C61 

40 

.   21 

.6-3:340 

1.57379 

.65313 

1.51946 

.65-343 

1.46.3-2(J 

.709.35 

1.40974 

39 

;  22 

.6-3-350 

1.57773 

.655.54 

1.51350 

.65.356 

1.46229 

.70979 

1.40S37 

33 

23 

.63121 

1.57676 

.65396 

1.51754 

.65429 

1.46137 

.710-23 

1.40300 

37 

24 

.6-3462 

1.57575 

.65933 

1.51653 

.63471 

1.46046 

.71066 

1.40714 

36 

25 

.6-3503 

1.57474 

.65950 

1.51562 

.65514 

1.4.5955 

.71110 

1.40627 

35 

26 

.6.3->14 

1.57372 

.66021 

1.51466 

.65557 

1.4-5564 

.71154 

1.40-510 

M 

27 

.63.534 

1.57271 

.60G63 

1.51370 

.63600 

1.45773 

.71193 

1.4W54 

.33 

28 

.63625 

1.57170 

.66105 

1.51275 

.63642 

1.45632 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179 

.63635 

1.4-5592 

.71235 

1.40231 

31 

30 

.63707 

1.56969 

.66159 

1.51034 

.63723 

1.45501 

.71329 

1.40195 

30 

31 

.63743 

1.56563 

.66230 

1.50933 

.63771 

1.4.5410 

.71373 

1.40109 

29 

32 

.63739 

1.56767 

.66272 

1.50393 

.63314 

1.45320 

.71417 

1.4'X)22 

28 

as 

.6:3330 

1.56667 

.66314 

1.50797 

.63357 

1.45229 

.71461 

1.39936 

27 

34 

.63371 

1.56566 

.66:356 

1.50702 

.63900 

1.45139 

.71505 

1.39350 

26 

35 

.63912 

1.56466 

.66393 

1.-50607 

.63942 

1.4-5049 

.71549 

1.39764 

25 

36 

.639.53 

1.56-366 

.66440 

1.50512 

.63985 

1.44953 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.664^2 

1.50417 

.69028 

1.44563 

.71637 

1.39-593 

23 

33 

.&4035 

1.56165 

.66-524 

1.50322 

.69071 

1.44773 

.71631 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50223 

.69114 

1.44633 

71725 

1.39421 

21 

40 

.&4117 

1.. 5.5966 

.66603 

1.50133 

.69157 

1.44.593 

71769 

1.39336 

20 

41 

.641-53 

1.-55566 

.66650 

1.50033 

.69200 

1.44503 

71813 

1.392.50 

19 

42 

.&4199 

1.55766 

.666.:.2 

1.49944 

.69-243 

1.44413 

.71857 

1.39165 

18 

43 

.64240 

1.55666 

.66734 

1.49549 

.69256 

1.44.3-29 

.71901 

1.39079 

17 

44 

.64231 

1.55;'567 

.66776 

1 .49755 

.69:329 

1.442:39 

.71946 

1.35994 

16 

45 

.643-22 

1.55467 

.66313 

1.49661 

.69:372 

1.44149 

.71990 

1.33909 

15 

46 

.61363 

1.55363 

.66360 

1.49566 

.69416 

1.44060 

.72034 

1.33824 

14 

47 

.64404 

1.. 5.5269 

.66902 

1 .49472 

.694-59 

1.43970 

.72073 

1.357.38 

13 

48 

.64446 

1.55170 

.66944 

1.49373 

.69-502 

1.43531 

.72122 

1.35653 

12 

49 

.64437 

1.5.5071 

.66936 

1.49-234 

.69-545 

1.43792 

.72167 

1.33563 

11 

50 

.64.528 

1.-54972 

.6702.3 

1.49190 

.69.533 

1.43703 

.72211 

1.. 33134 

10 

51 

.64-569 

1.54373 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.33399 

9 

52 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43:325 

.72299 

1.33314 

8 

53 

.64652 

1.54675 

.67155 

1.45909 

.69718 

1.434-36 

.72.344 

1.33229 

7 

54 

.64693 

1.54.576 

.67197 

1.43316 

.69761 

1.4.3347 

.72.358 

1.38145 

6 

55 

.64734 

1.54478 

.672-39 

1.48722 

.69504 

1.43253 

.72432 

1. 33060 

5 

56 

.64775 

1.54379 

.67232 

1.48629 

.69347 

1.43169 

.72477 

1.. 37976 

4 

57 

.64SI7 

1.54231 

.67324 

1.435.36 

.69591 

1.4.30SO 

.72-521 

1.37891 

3 

53 

.643.53 

1.54133 

.67.366 

1.45442 

.69934 

1.42992 

.72565 

1.37507 

2 

59 

.64399 

1.54035 

.67409 

1.43^9 

.69977 

1.42903 

.72610 

1.37722 

1 

j   60 
f  M. 

1 

.64941 

1.. 5.3956 

.67451 

1.48-256 
Tang.     ( 

.7(021 

1.42315 
Tang. 

.726.34 

1.37638 

0 
M. 

Gotang. 

Tang. 

Cotang. 

Cotang. 

Cotang. 
5' 

Tang. 

to 

5 

r= 

5 

6=             1 

5 

53 

TABLE  XV.      NATURAL  TANGENTS  AND  COTANGENTS. 


239 


36 


M.'  Tang.     Cotang. 

1 

21 

3| 
41 
5  i 
6i 

rr 

I 

81 

^1 

10 

11 1 

12' 

13  i 

14  1 


37^ 


10  ' 

16! 

17  i 

18! 

19 

20 

21 

22 

2;i 

2-1 

25 

26 

27 

23 

29 

30 


.72654  , 

.72699  I 

.72743  I 

.72783 

.72S32 

.72S77 

.72921 

.72966 

.73010 

.7.3055 

.73100 

.73144 

.731 S9 

.73-234 

.73278 

.73323 

.7336S 

.73413 

.73457 

.73502 

.73.547 

.73592 

\   .73637 

\   .73631 

;'.  73726 

;  .73771 

\  .7.3S16 

.7.3S61 

.73906 

.73951 

.73996 

31  .74041 

32  I  .74056 
331  .74131 
34  I  .74176 

.74221 
.74267 
.74312 
.74357 
.74402 
.74447 
.74492 


Tang.     Cotang. 


35 
36 
37 
33 
39 
40 
41 
42 
43 
44 
45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

53 

59 

60 


.74533 
.74-533 

.74623 
.74674 

.74719 

.74764 

.74510 

.74355 

.74900 

.74946 

.74991 

.75037 

.75032 

.75123 

.75173 

.75219 

.75264 

.75310 

.75355 


1 .37633 

1.37554 

1.37470 

1.373S6 

1.37302 

1.3721S 

1.37134 

1.370^50 

1.36967 

1.36^33 

1.36-00 

1.. 367 1 6 

1.36633 

1.36549 

1.3&166 

1.36333 

1.36300 

1.36217 

1.36134 

1.36051 

1.3.5963 

1.3533^5 

1.35302 

1.35719 

1.35637 

1.3.5554 

1.35472 

1.353=9 

1.35307 

1.3.5-224 

1.35142 

1.3.5060 

1.34973 

1.34396 

1.:34314 

1.34732 

1.34650 

1.34563 

1.34437 

1.34405 

1.34323 

1.34242 

1.34160 

l.:31079 

1.33993 

1.33916 


M.;Dotang. 


1.33335 

1.337.54 

1.33673 

1.33.592 

1.33511 

1.33430 

1.33349 

1.3.3263 

1.33137 

1.33107 

1.33026 

1.32946 

1.32365 

1.32735 

1.32704 


.75401 

.7;5447 

.75492 

.75533 

.75534 

.75629 

.75675 

.75721_ 

.75767 

.75312 

.75353 

.75904 

.75950 

.75996 

.76042 

.76033 

.76134 

.76130 

.76226 

.76272 

.76313 

.76364 

.76410 

.76456 

,76502 

.76543 

.?6594 

.76640 

.76636 

.76733 

.76779 

.76325 

.76371 

.76913 

.76964 

.77010 

.770-57 

.77103 

.77149 

.77196 

.77242 

.77239 

.77335 

.773S2 

.77428 

.77475 

.77521 

.77563 

.77615 

.77661 

.77703 

.77754 

.77301 

.77343 

.77395 

.77W1 

.77933 

.73035 

.73082 

.73129 


38^ 


Tang. 


1.32704 

1.32624 

1.32^544 

1.32464 

1.32.334 

1.32.304 

1.32224 

1.32144 

1.32064 

1  31934 

1.3190-1 

1.31S25 

1.31745 

1.31666 

1.31556 

1.31507 

1.31427 

1.31343 

1.31269 

1.31190 

1.31110 

1.31031 

1.30952 

1.30373 

1.30795 

1.30716 

1.30637 

1.30553 

1.30430 

I.3<3401 

1.30323 

1.30244 

1.30166 

1.30037 

1.30009 

1.29931 

1.293-53 

1.29775 

1.29696 

1.29613 

1.29.541 

1.29463 

1.29335 

1.29307 

1.29229 

1.29152 


Cotang. 


39c 


" 


.73129 

.73175 

.78222 

.78269 

.78316 

.76363 

.73410 

.78457 

.76504 

.73-551 

.78598 

.78645 

.78692 

.78739 

.78786 

.78834 

.78831 

.78928 

.78975 

.79022 

.79070 

.79117 

.79164 

.79212 

.79259 

.79-306  ! 

.79354 

.79401 

.79449 

.79496 

.79544 

.79591 

.796:39 

.79636 

.79734 

.79781 

.79329 

.79377 

.79924 

.79972 

.80020 

.80067 

.80115 

.80163 

.80211 

.80253 


1.27994 

1.27917 

1.27541 

1.27764 

1.27633 

1.27611 

1.27535 

1.274-53 

1.27,332 

1.27396 

1.27230 

1.27  i53 

1.27077 

1.27C01 

1.26925 

1.26549 


Tang.     Cotang.  ^M. 


.60973 

.Slu27 

.51075 

.51123 

.81171 

.81220 

.81263 

.81316 

.81364 

.81413 

.81461 

.81510 

.81553 

.81606 

.81655 

.81703 


Tang. 


53^ 


1.29074 

1.2>997 

1.23919 

1.23342 

1.23764 

1.2.5637 

1.25610 

1.23533 

1.23456 

1.23379 

1.28302 

1.23-225 

1.23143 

1.23071 

1 .27994 


Cotang.      Tang. 

53= 


1.26774 

1.26693 

1.266-22 

1.26546 

1.26471 

1.26395 

1.26319 

1.26-244  I 

1.26169 

1.26093 

1.26013 

1.25943 

1.25567 

1.25792 

1.25717 

1.25642 
1.25567 
1.2-5492 
1.2.5417 
1.25343 
1 .2-5-263 
1.2:5193 
1.25113 
1.25044 
1.24969 
1.24595 
1.245-20 
1.24746 
1.24672 
1.24597 

1.24523 

1.24449 

1.24375 

1.24301 

1.24227 

1.24153 

1.24079 

1.24005 

1.2-3931 

1.23553 

1.23784 

1.23710 

1.23637 

1.23563 

1.23490 

Cotang.      Tang. 
513 


.80306 

.80354 

.80402 

.80450 

.50493 

.50546 

.80594 

.80642 

.50690 

.80733 

.80736 

.50334 

.80552 

.80930 

.80973 


.81752 

.61300 

.81849 

.51593 

.31946 

.51995 

.3-2044 

.52092 

.82141 

.52190 

.5-22.33 

.R2287 

.5^:336 

.52335 

.82434 

.52453 

.5-2.531 

.5-2530 

.32629 

.5-2678 

.52727 

.3-2776 

.82325 

.32374 

.52923 

.5-2972 

.53022 

.83071 

,53120 

.53169 

.83215 

.53268 

.83317 

.83366 

.83415 

.83465 

.53514 

.53564 

.83613 

.53662 

.33712 

.83761 

.8331 1 

.83560 

.33910 


1.-23490 

1.23416 

1.2-3343 

1.23270 

1.23196 

1.23123 

1.2.30.50 

1.22977 

1.2-2904 

1.22531 

1.-2-2753 

1.2-2635 

1.-2-2612 

1.2-2539 

1.-2-2467 

1.-2-23W 

1.22321 

1.22-249 

1.22176 

1.22104 

1.22031 

1.21S59 

1.21586 

1.21814 

1.21742 

1.21670 

1.21593 

1.215-26 

1.21454 

1.21352 

1.21310 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

43 

47 

46 

45 

44 

43 

42 

41 

40 

39 

33 

37 

36 

35 

34 

33 

32 

31 

30 


1.21-2.33 

1.21166 

1.21094 

1.210-23 

1.20951 

1.20579 

1.20503 

1. -20736 

1.20665 

;. 20593 

1.20522 

1.20451 

1.20379 

1.20308 

1.-20-237 

1.20166 
1.20095 
1.20024 
-..19953 
1.19582 
1.19511 
1.19740 
1.19669 
1.19.599 
1.19523 
1.19457 
1  19387 
1.19316 
1.19-246 
1  19175 


29 

23 

27 

-26 

25 

24 

23 

22 

21 

20 
19 
18 
17 
16 
15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cotang.^    Tang.     M. 
503  '1 


240       TABLE  XV.      NATURAL  TANGENTS  AND  COTANGENTS. 


M. 

0 

4:03 

4:10                 1 

4: 

20 

433     1 

M. 

60 

1 

! 
1 

1 

Tang. 

.S39I0 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 
1.11061 

Tang. 

Cotang. 

1.19175 

.86929 

1.15037 

.90040 

.93252 

1.072.37 

1 

.83960 

1.19105 

.S6930 

1.14969 

.90093 

1.10996 

.93306 

1.07174 

59 

( 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

1.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87032 

1.14334 

.90199 

1.10367 

.93415 

1.07049 

57 

4 

.84103 

1.18394 

.87133 

1.14767 

.90251 

1.10302 

.93469 

1.06937 

56 

5 

.84153 

1.18324 

.871.34 

1.14699 

.90304 

1.10737 

.93524 

1.06925 

55 

j 

6 

.84203 

1.18754 

.87236 

1.14632 

.90357 

1.10672 

.93578 

1.06362 

54 

1 

7 

.84258 

1.1 8634 

.b72S7 

1.14565 

.90410 

1.10607 

.93633 

1.06300 

53 

1 

8 

.84307 

1.18614 

.87333 

1.14493 

.90463 

1.10543 

.93638 

1.06733 

52 

1 

9 

.84357 

1.13544 

.87339 

1.14430 

.90516 

1.10473 

.93742 

1.06676 

51 

1 

10 

.84407 

1.13474 

.87441 

1.14363 

.90569 

1.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.90621 

1.10349 

.93352 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

1.10235 

.93906 

1.06439 

48 

13 

.84556 

1.18264 

.87595 

1.14162 

.90727 

1.10220 

.93961 

1.06427 

47 

14 

.84606 

1.13194 

.87646 

1.14095 

.90731 

1.10156 

.94016 

1.06365 

46 

15 

.84656 

1.13125 

.87693 

1.14023 

.90334 

1.10091 

.94071 

1.06.303 

45 

16 

.84706 

1.13055 

.87749 

1.13961 

.90337 

1.10027 

.94125 

1.06241 

44 

17 

.84756 

1.17936 

.87801 

1.13394 

.90940 

1.09963 

.94180 

1.06179 

43 

18 

.84306 

1.17916 

.87352 

1.13323 

.90993 

1.09399 

.94235 

1.06117 

42 

19 

.84356 

1.17346 

.87904 

1.13761 

.91046 

1.09834 

.94290 

1.06056 

41 

20 

.84906 

1.17777 

.87955 

1.13694 

.91099 

1.09770 

.94345 

1.05994 

40 

21 

.84956 

1.17703 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.059.32 

39 

22 

.85006 

1.17633 

.83059 

1.13561 

.91206 

1.09642 

.94455 

1.0.5370 

33 

23 

.85057 

1.17569 

.83110 

1.13494 

.91259 

1.09573 

.94510 

1.0.5309 

37 

24 

.85107 

1.17500 

.83162 

1.13423 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.174.30 

.83214 

1.13361 

.91366 

1.09450 

.94620 

1.05635 

35 

26 

.8.5207 

1.17351 

.83265 

1.13295 

.91419 

1.09336 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.83317 

1.13223 

.91473 

1.09322 

.94731 

1.05562 

33 

23 

.85303 

1.17223 

.83369 

1.13162 

.91526 

1.09253 

.94736 

1.05501 

32 

29 

.85353 

1.17154 

.83421 

1.13096 

.91530 

1.09195 

.94341 

1.05439 

31 

30 

.85403 

1.17035 

.83473 

1.13029 

.91633 

1.09131 

.94396 

1.05378 

30 

31 

.Si>453 

1.17016 

.83.524 

1.12963 

.91637 

1.09067 

.94952 

1.05317 

29 

32 

.85509 

1.16947 

.83576 

1.12397 

.91740 

1.09003 

.95007 

1.05255 

23 

33 

.8.5559 

1.16378 

.88623 

1.12331 

.91794 

1.08940 

.95062 

1.05194 

27 

34 

.85609 

1.16309 

.83630 

1.12765 

.91347 

1.03876 

.95113 

1.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

1.03813 

.95173 

1.05072 

25 

36 

.85710 

1.16672 

.83784 

1.12633 

.91955 

1.08749 

.9.5229 

1.05010 

24 

37 

.85761 

1.16603 

.88336 

1.12567 

.92003 

1.03636 

.95234 

1.04949 

23 

38 

.85311 

1.16535 

.83333 

1.12501 

.92062 

1.0S622 

.9.5340 

1.04333 

22 

39 

.85362 

1.16466 

.83940 

1.124.35 

.92116 

1.03-559 

.95395 

1.04327 

21 

40 

.85912 

1.16.393 

.83992 

1.12369 

.92170 

1.03496 

.95451 

1.04766 

20 

41 

.85963 

1.16.329 

.89045 

1.12303 

.92224 

1.03432 

.95506 

1.04705 

19 

42 

.86014 

1.16261 

.89097 

1.122.33 

.92277 

1.03369 

.95562 

1.04644 

18 

43 

.86061 

1.16192 

.89149 

1.12172 

.92331 

1.03306 

.95618 

1.04.533 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92335 

1.03243 

.95673 

1.04.522 

16 

45 

.86166 

1.16056 

.892.53 

1.12041 

.92439 

1.03179 

.95729 

1.04461 

15 

46 

.86216 

1.15937 

.89306 

1.11975 

.92493 

1.03116 

95785 

1.04401 

14 

47 

.86267 

1.1.5919 

.893.53 

1.11909 

.92547 

1.03053 

.95341 

1.04340 

13 

48 

.86318 

1.1.5351 

.89410 

1.11844 

.92601 

1.07990 

.95397 

1.04279 

12 

49 

.86363 

1.1.5733 

.89463 

1.11778 

.92655 

1.07927 

95952 

]. 04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96003 

1.041.53 

10 

51 

.86470 

1.15647 

.89.567 

1.11643 

.92763 

1.07301 

.96064 

1.04097 

9 

52 

.86521 

1.15.579 

.89620 

1.11532 

.92817 

1.07733 

.96120 

1.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92372 

1.07676 

.96176 

1.03976 

7 

54 

.86623 

1.1.5443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

1.0.3915 

6 

55 

.86674 

1.15375 

.89777 

1.11.337 

.92930 

1.07550 

.96238 

1.03355 

5 

56 

.86725 

1.15303 

.89330 

1.11.321 

.93034 

1.07437 

.96:344 

1.03794 

4 

57 

.86776 

1.15240 

.89SS3 

1.112.56 

.93038 

1.07425 

.96400 

1.03734 

3 

58 

.36327 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.964.57 

1.03674 

2 

59 

.86378 

1.15104 

.89938 

1.11126 

.93197 

1.07299 

.96513 

1.0-3613 

1 

60 
M. 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

0 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

Cotang. 

Tang. 

4 

9= 

4 

:8  = 

4 

.70 

463 

TABLE  XV.      NATURAL  TANGENTS  AND  COTANGENTS.       241 


M. 

(J 

440     1 

M. 

fiO 

M. 

20 

440 

M. 

40 

M. 

40 

440 

20 

Tang. 

Cotang. 

Tang. 

Cotang. 
1.02355 

Tang. 

Cotang. 

.96569 

1.03553 

.97700 

.98843 

1.01170 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.98901 

1.01112 

19 

9 

.96631 

1.03133 

58 

22 

.97813 

1.02236 

38 

42 

.989.58 

1.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

1.00994 

1/ 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

36 

44 

.99073 

1.00935 

16 

5 

.96350 

1.03252 

55 

25 

.97934 

1.02057 

35 

45 

.99131 

1.00376 

15 

fi 

.96907 

1.03192 

54 

2(5 

.93041 

1.01998 

34 

46 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.93093 

1.01939 

33 

47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

28 

.93155 

1.01879 

32 

48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.93213 

1.01820 

31 

49 

.99362 

1.00642 

11 

10 

.97ia3 

1.02952 

50 

30 

.93270 

1.01761 

30 

50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

1.01702 

29 

51 

.99478 

1.00525 

9 

P 

.97246 

1.02832 

48 

32 

.93334 

1.01642 

28 

52 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

33 

.93441 

1.01533 

27 

53 

.99594 

1.00403 

"i 

14 

.97359 

1.02713 

46 

34 

.93499 

1.01524 

26 

54 

.99652 

1.00350 

6 

15 

.974)3 

1.02653 

45 

ai 

.93556 

1.01465 

25 

55 

.99710 

1.00291 

b 

16 

.974: 2 

1.02593 

44 

36 

.93613 

1.01406 

24 

56 

,99763 

1.00233 

4 

17 

.97529 

1.02533 

43 

37 

.93671 

1.01347 

23 

57 

.99326 

1.00175 

3 

18 

.97536 

1.02474 

42 

38 

.93728 

1,01283 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 
M. 

.97700 

1.02355 

40 
M. 

40 
M. 

.93343 

1.01170 

20 
M. 

60 
M. 

1.00000 

1.00000 

0 
M. 

Cotang. 

Tang. 

Coteing. 

Tang. 

Cotang. 

Tang. 

453 

450 

450 

V 


242       TABLE  XVI.      RISE  PER  MILE  OF  VARIOUS  GRADES. 


TABLE    XVI. 


RISE   PER  MILE   OE  VARIOUS   GRADES. 


Grade 

per 
Htatioa. 

Rise  per 

Mile- 

Grade 
per 

Station. 

Rise  per 
Mile. 

Grade 

per 
Station. 

Rise  per 
Mile. 

Grade 

per 
Station. 

Rise  per 
Mile. 

.01 

.523 

.41 

21.643 

.81 

42.763 

1.21 

63.838 

.02 

1.0.56 

.42 

22.176 

.52 

43.296 

1.22 

64.416 

.03 

1.5S4 

.43 

22.701 

.83 

43..y2l 

1.23 

64.944 

.04 

2.112 

.44 

23.2.32 

..S4 

44.3.52 

1.24 

65.472 

.05 

2.640 

.45 

23.760 

.85 

44.850 

1.25 

66.000 

.06 

3.163 

.46 

24.233 

.86 

45.403 

1.26 

66.523      , 

.07 

3.6S6 

.47 

24.816 

.37 

45.936 

1.27 

67.056 

.OS 

4.224 

.43 

2-5.344 

.83 

46.464 

1.23 

67.534 

.09 

4.752 

.49 

25.872 

.89 

46.992 

1.29 

63.112 

.10 

5.280 

.50 

26.400 

.90 

47.520 

1.30 

65.640 

.11 

5.803 

.51 

26.923 

.91 

48.043 

1.31 

69.163 

.12 

6.3.36 

.52 

27.4-56 

.92 

43.576 

1.32 

69.696 

.1.3 

6.S64 

.53 

27.934 

.93 

49.104 

1.33 

70.224 

.14 

7.392 

.54 

23.512 

.94 

49.632 

1.34 

70.752 

.15 

7.920 

.55 

29.040 

.95 

5OI60 

1.35 

71.230 

.16 

8.443 

.56 

29.563 

.96 

50.683 

1.36 

71.808 

.17 

8.976 

.57 

30096 

.97 

51.216 

1.37 

72.336 

.13 

9.504 

.53 

30.624 

.93 

51.744 

1.33 

72.S64 

.19 

10.032 

.59 

31.152 

.99 

52.272 

1.39 

73.392 

.20 

10.  .560 

.60 

31.6S0 

1.00 

52.800 

1.40 

73.920 

.21 

11.083 

.61 

32.203 

l.ni 

53.323 

1.41 

74.443 

.22 

11.616 

.62 

32.738 

IM 

53.8.56 

1.42 

74.976 

.23 

12.144 

.63 

33.264 

1.03 

54.354 

1.43 

75.. 504 

.24 

12.672 

.64 

33.792 

1.04 

54.912 

1.44 

76.0.32 

.25 

13.200 

.65 

34.320 

1.05 

55.440 

1.45 

76.560 

.26 

13.723 

.66 

34.S43 

1.06 

55.963 

1.46 

77.038 

.27 

14.2.56 

.67 

35.376 

1.07 

56.496 

1.47 

77.616 

.23 

14.784 

.63 

35.904 

1.03 

57.024 

1.43 

78.144 

.29 

15.312 

.69 

36.432 

1.09 

57.552 

1.49 

78.672 

.30 

15.840 

.70 

36.960 

1.10 

53.030 

1.50 

79.200 

.31 

16.363 

.71 

37.483 

l.Il 

53.608 

1.51 

79.723 

.32 

16.896 

.72 

33.016 

1.12 

59.1.36 

1.52 

80.2.56 

.33 

17.424 

.73 

33.544 

1.13 

59.664 

1.53 

80.784 

.34 

17.952 

.74 

39.072 

1.14 

60192 

1.54 

81.312 

.3.5 

18.450 

.75 

39.600 

1.15 

60.720 

1.55 

81.840 

.36 

19.003 

.76 

40123 

1.16 

61.243 

1.56 

82.363 

.37 

19.536 

.77 

40.656 

1.17 

61.776 

1.57 

82.896 

.33 

20.0&4 

.78 

41.184 

1.18 

62.304 

1.58 

83.424 

.39 

20.592 

.79 

41.712 

1.19 

62.832 

1.59 

a3.952 

.40 

21.120 

.80 

42.240 

1.20 

63.360 

1.60 

&i.480 

TABLE  XVI.      RISE    PER  MILE  OF  VARIOUS  GRADES.       243 


Grade 

Rise  per 

Grade 

Rise  per 

Grade 

Rise  per 

Grade 

Rise  per 

per 
Station. 

Mile. 

per 
Station. 

Mile. 

per 
Station. 

Mile. 

per 
Station. 

Mile. 

1.61 

S5.003 

1.81 

95.563 

2.10 

110.880 

4.10 

2I6.4S0 

1.62 

65.536 

1.82 

96.096 

2.20 

116.160 

4.20 

221.760 

1.63 

86.064 

1.S3 

96.624 

2.30 

121.440 

4.30 

227.040 

1.64 

86.592 

1.84 

97.152 

2.40 

126.720 

4.40 

232.320 

1.6.5 

87.120 

1.85 

97.630 

2.50 

132.000 

4.50 

237.600 

1.66 

87.643 

1.86 

98.208 

2.60 

137.280 

4.60 

242.880 

1.67 

88.176 

1.87 

93.736 

2.70 

142.560 

4.70 

243.160 

1.63 

88.704 

I.S8 

99.264 

2.80 

147.840 

4.80 

253.440 

1.69 

89.232 

1.89 

99.792 

2.90 

153.120 

4.90 

253.720 

1.70 

89.760 

1.90 

100.320 

3.00 

153.400 

5.00 

264.000 

1.71 

90.233 

1.91 

100.843 

3.10 

163.680 

5.10 

269.230 

1.72 

90.816 

1.92 

101.376 

3.20 

163.960 

5.20 

274.560 

1.73 

91.344 

1.93 

101.904 

3.30 

174.240 

5.30 

279.840 

1.74 

91.872 

1.94 

102.432 

3.40 

179.520 

5.40 

235.120 

1.75 

92.400 

1.95 

102.960 

3  50 

184.800 

5.50 

290.400 

1.76 

92.923 

1.96 

103.483 

3.60 

190.080 

5.60 

295.630 

1.77 

93.456 

1.97 

104.016 

3.70 

195.360 

5.70 

300.960 

1.73 

93.934 

1.93 

104.544 

3.80 

200.640 

5.80 

306.240 

1.79 

94.512 

1.99 

105.072 

3.90 

205.920 

5.90 

311.520 

l.SO 

95.040 

2.00 

105.600 

4.00 

211.200 

6.00 

316.800 

THE  mm 


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UNIVERSITY  OF  ILLINOIS-URBANA 


3  0112  084205183 


